Chuck Peterson's Problem Solving Page
Guess and Check Problems
Look for a Pattern
Make a Systematic List
Make a Drawing or Model
Simplify the Problem
Look for a Pattern
Make a Systematic List
Make a Drawing or Model
Simplify the Problem
Guess and Check is a useful strategy for certain types of problems. It allows one to make a guess, check against the relevant facts in the problem, and refining the guess to get closer to the correct answer.
Example: Matthew is 12 years old and his mother is three times as old. How many years must pass before his mother is twice as old? Is the answer 4 years? No, because in 4 years, she would be 40 years old and he would be 16 and 40 is not twice 16. Refine your guess. Is 20 years correct? No, she would be 56 and he would be 32. 56 is not twice 32. Refine your guess ............ and check.
Day 1 - Barbara has exactly $2.00 in nickels and dimes. She has twice as many dimes as nickels. How many of each does she have?
Day 2 - Mary bought a scarf for $5.00, spent 1/2 of her remaining money on jogging shoes, bought lunch for $2.00, then spent 1/2 of her remaining money on a CD. She had $10.00 left. How much did she start with?
Day 3 - (a) I'm thinking of a number. If you multiply it by 3, then subtract 5 and finally add 10, you get 20. What number am I thinking of? (b) I'm thinking of a number. If you multiply its square by 3 and then add 9, you get 117. What is that number? (c) I'm thinking of a number. If you subtract 4 from the number, then multiply the result by 3 and then add 5, you get 26. What is that number?
Day 4 - Each check I write costs 10 cents. I also have to pay a flat fee of 25 cents per month. My bank sent me a letter saying the fees were going to change. The new fee is 8 cents for each check and a flat fee of 50 cents/month. The bank said this would be cheaper for me because of the number of checks I write. What is the least number of checks I must write to make the new fees cheaper?
Day 5 - There are two rectangles with whole-number side lengths for which the perimeter is the same number as the area. Find both rectangles.
Day 26 - In a geometry course the grade is based on six tests, each worth 100 points. W. Orrier has an average of 88.5 on his first four tests. What is the lowest average he could obtain on his next two tests and still receive an average of 90 or better?
Day 27 - A Super Notebook was on sale last week at 15 % off the regular price. Then an additional 10 % of the sale price was deducted to give a final super sale price of $ 25.09. What was the regular price of the notebook?
Day 28 - Pythagoras discovered amicable or "friendly" numbers. Two positive integers are amicable if each is the sum of the proper divisors of the other. 284 is amicable with 220. What number is amicable to 1184?
Day 29 - [This one seems to be missing information! If you know a good modification, please email octm.webmaster@gmail.com.] If 100 bushels of corn is distributed so that each man receives 3 bushels, each woman 2 bushels, and each child 1/2 bushel, how many men, women and children are there?
Day 30 - My license tag is a three-digit number. The product of the digits is 216, and their sum is 19 and the digits appear in ascending order. Find the license plate number.
Day 51 - A two-digit number is divided by the sum of its digits. What is the largest attainable remainder?
Day 53 - Remove six of these 24 line segments to leave three squares.
Day 54 - A girl had her monthly allowance doubled, next received an additional $3 increase, and then had her allowance cut in half. How much more or less is her present allowance compared with her original allowance?
Day 55 - A stock market analyst sold a monthly newsletter to 500 subscribers at a price of $10 each. She discovered that for each $0.25 increase in the monthly price of the newsletter, she would lose 2 subscriptions. For what price should she sell each issue to bring in the greatest total monthly income?
Day 76 - Using four 4's and no other number and any mathematics operation symbol(s), write an expression whose value is 19. (For example, 44/4 + 4 = 15.)
Day 77 - Find three consecutive odd integers whose sum is -3.
Day 78 - Write the number 1 using each of the nine digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 only once.
Day 79 - Place parenthesis to make this statement true: 9 x 5 + 2 - 8 x 3 + 1 = 22
Day 80 - How can you place 21 marbles in four boxes so that each box contains an odd number of marbles?
Day 101 - In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. How many cows are there?
Day 103 - Determine all the values of x and y so that x/y, x times y, and x - y are all equal.
Day 104 - Place the numbers 1 through 10 in the blanks so that any number is the absolute value of the two numbers directly above it.
Example: Matthew is 12 years old and his mother is three times as old. How many years must pass before his mother is twice as old? Is the answer 4 years? No, because in 4 years, she would be 40 years old and he would be 16 and 40 is not twice 16. Refine your guess. Is 20 years correct? No, she would be 56 and he would be 32. 56 is not twice 32. Refine your guess ............ and check.
Day 1 - Barbara has exactly $2.00 in nickels and dimes. She has twice as many dimes as nickels. How many of each does she have?
Day 2 - Mary bought a scarf for $5.00, spent 1/2 of her remaining money on jogging shoes, bought lunch for $2.00, then spent 1/2 of her remaining money on a CD. She had $10.00 left. How much did she start with?
Day 3 - (a) I'm thinking of a number. If you multiply it by 3, then subtract 5 and finally add 10, you get 20. What number am I thinking of? (b) I'm thinking of a number. If you multiply its square by 3 and then add 9, you get 117. What is that number? (c) I'm thinking of a number. If you subtract 4 from the number, then multiply the result by 3 and then add 5, you get 26. What is that number?
Day 4 - Each check I write costs 10 cents. I also have to pay a flat fee of 25 cents per month. My bank sent me a letter saying the fees were going to change. The new fee is 8 cents for each check and a flat fee of 50 cents/month. The bank said this would be cheaper for me because of the number of checks I write. What is the least number of checks I must write to make the new fees cheaper?
Day 5 - There are two rectangles with whole-number side lengths for which the perimeter is the same number as the area. Find both rectangles.
Day 26 - In a geometry course the grade is based on six tests, each worth 100 points. W. Orrier has an average of 88.5 on his first four tests. What is the lowest average he could obtain on his next two tests and still receive an average of 90 or better?
Day 27 - A Super Notebook was on sale last week at 15 % off the regular price. Then an additional 10 % of the sale price was deducted to give a final super sale price of $ 25.09. What was the regular price of the notebook?
Day 28 - Pythagoras discovered amicable or "friendly" numbers. Two positive integers are amicable if each is the sum of the proper divisors of the other. 284 is amicable with 220. What number is amicable to 1184?
Day 29 - [This one seems to be missing information! If you know a good modification, please email octm.webmaster@gmail.com.] If 100 bushels of corn is distributed so that each man receives 3 bushels, each woman 2 bushels, and each child 1/2 bushel, how many men, women and children are there?
Day 30 - My license tag is a three-digit number. The product of the digits is 216, and their sum is 19 and the digits appear in ascending order. Find the license plate number.
Day 51 - A two-digit number is divided by the sum of its digits. What is the largest attainable remainder?
Day 53 - Remove six of these 24 line segments to leave three squares.
Day 54 - A girl had her monthly allowance doubled, next received an additional $3 increase, and then had her allowance cut in half. How much more or less is her present allowance compared with her original allowance?
Day 55 - A stock market analyst sold a monthly newsletter to 500 subscribers at a price of $10 each. She discovered that for each $0.25 increase in the monthly price of the newsletter, she would lose 2 subscriptions. For what price should she sell each issue to bring in the greatest total monthly income?
Day 76 - Using four 4's and no other number and any mathematics operation symbol(s), write an expression whose value is 19. (For example, 44/4 + 4 = 15.)
Day 77 - Find three consecutive odd integers whose sum is -3.
Day 78 - Write the number 1 using each of the nine digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 only once.
Day 79 - Place parenthesis to make this statement true: 9 x 5 + 2 - 8 x 3 + 1 = 22
Day 80 - How can you place 21 marbles in four boxes so that each box contains an odd number of marbles?
Day 101 - In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. How many cows are there?
Day 103 - Determine all the values of x and y so that x/y, x times y, and x - y are all equal.
Day 104 - Place the numbers 1 through 10 in the blanks so that any number is the absolute value of the two numbers directly above it.
Day 105 - Using the digits in 1996 and any mathematical operation, generate the numbers 1 through 20. For example, 19 - 9 + 6 = 16.
Day 126 - The operation @ is defined as a @ b = a^2 + 3b. Find four pairs of natural numbers such that a @ b = 37
Day 127 - How can you have $2 in nickels, dimes, and quarters, with the same number of each coin?
Day 128 - Place the numbers 2, 3, 4, 5, 6, 7, 8, 9 and 10 in the boxes so that the sum of the numbers in the boxes of each of the four circles is 27.
Day 126 - The operation @ is defined as a @ b = a^2 + 3b. Find four pairs of natural numbers such that a @ b = 37
Day 127 - How can you have $2 in nickels, dimes, and quarters, with the same number of each coin?
Day 128 - Place the numbers 2, 3, 4, 5, 6, 7, 8, 9 and 10 in the boxes so that the sum of the numbers in the boxes of each of the four circles is 27.
Day 129 - A mathematics teacher's son was born in the 1900s and will be x years old in the year x^2. In what year was he born?
Day 130 - Use exactly three 3's and any nonnumerical mathematical symbols ( i.e. +, -, x, /, sq. rt., etc.) to write expressions that equal each of the numbers one through ten inclusive.
Day 130 - Use exactly three 3's and any nonnumerical mathematical symbols ( i.e. +, -, x, /, sq. rt., etc.) to write expressions that equal each of the numbers one through ten inclusive.
Much of mathematics is looking for and applying useful patterns.
Example: What is the sum of the first 320 odd numbers? By looking at the pattern for the sum of the first two odd numbers (1+3=4), the first three odd numbers (1+3+5=9), the first four odd numbers ( 1+3+5+7=16) and so on, the sum of the first 320 odd numbers is quickly solved. (320^2)
Day 6 - What is the sum of the first 4 odd numbers? 5 odd numbers? 7 odd numbers? 10 odd numbers? 2356 odd numbers?
Day 7 - Find the pattern. Fill in the blanks. (a) 1 , 4 , 9 , 16, 25, __, __, __ (b) 3, 4, 7, 11, 18, 29, __, __, __ (c) 5, 10, 9, 18, 17, 34, 33, __, __ (d) 1, 2, 6, 24, 120, __, __, __ (e) 77, 49, 36, 18, __
Day 8 - Find these products: 7 x 9, 77 x 99, 777 x 999, Predict the product for 77,777 x 99999. What two numbers give a product of 77,762,223?
Day 10 - For the first 8 powers of 2, find the units digit. What will the units digit be for 2^12? 2^36? 2^9? 2^25? 2^39? 2^42?
Day 31 - Determine the number of integers between 1 and 1000 that contain at least one 2 but no 3.
Day 32 - How many numbers less than 124 are divisible by 2, 3 and 5?
Day 33 - Can you find a number that fits this pattern? 3600 , 1800, 600, 150, __?__
Day 34 - If you added 6 rows to the bottom of this picture, how many small triangles would you have altogether?
Example: What is the sum of the first 320 odd numbers? By looking at the pattern for the sum of the first two odd numbers (1+3=4), the first three odd numbers (1+3+5=9), the first four odd numbers ( 1+3+5+7=16) and so on, the sum of the first 320 odd numbers is quickly solved. (320^2)
Day 6 - What is the sum of the first 4 odd numbers? 5 odd numbers? 7 odd numbers? 10 odd numbers? 2356 odd numbers?
Day 7 - Find the pattern. Fill in the blanks. (a) 1 , 4 , 9 , 16, 25, __, __, __ (b) 3, 4, 7, 11, 18, 29, __, __, __ (c) 5, 10, 9, 18, 17, 34, 33, __, __ (d) 1, 2, 6, 24, 120, __, __, __ (e) 77, 49, 36, 18, __
Day 8 - Find these products: 7 x 9, 77 x 99, 777 x 999, Predict the product for 77,777 x 99999. What two numbers give a product of 77,762,223?
Day 10 - For the first 8 powers of 2, find the units digit. What will the units digit be for 2^12? 2^36? 2^9? 2^25? 2^39? 2^42?
Day 31 - Determine the number of integers between 1 and 1000 that contain at least one 2 but no 3.
Day 32 - How many numbers less than 124 are divisible by 2, 3 and 5?
Day 33 - Can you find a number that fits this pattern? 3600 , 1800, 600, 150, __?__
Day 34 - If you added 6 rows to the bottom of this picture, how many small triangles would you have altogether?
Day 35 - 1990 - 1980 + 1970 - 1960 + ..... - 20 + 10 = ?
Day 56 - Find the following sum: 1/(1x2) +1/(2x3) + 1/(3x4) + ..... + 1/(99x100)
Day 57 - What is the units digit of this sum? 1! + 2! + 3! + ...... + 14! + 15!
Day 58 - How many squares are contained in a 5 x 5 square grid?
Day 59 - Express as a single fraction in lowest terms: 1/(1x2) + 1/(2x3) + 1/(3x4) + ..... + 1/(9x10)
Day 60 - Find a possible next number for this sequence: 3 , 7, 16, 32, 57. 93, ___
Day 81 - What is the ones digit of 3^1992?
Day 82 - A town has a population of 300,000 with an annual growth rate of 4.5%. At that rate, in how many years will the population be 500,000?
Day 83 - What is a + b ?
Day 56 - Find the following sum: 1/(1x2) +1/(2x3) + 1/(3x4) + ..... + 1/(99x100)
Day 57 - What is the units digit of this sum? 1! + 2! + 3! + ...... + 14! + 15!
Day 58 - How many squares are contained in a 5 x 5 square grid?
Day 59 - Express as a single fraction in lowest terms: 1/(1x2) + 1/(2x3) + 1/(3x4) + ..... + 1/(9x10)
Day 60 - Find a possible next number for this sequence: 3 , 7, 16, 32, 57. 93, ___
Day 81 - What is the ones digit of 3^1992?
Day 82 - A town has a population of 300,000 with an annual growth rate of 4.5%. At that rate, in how many years will the population be 500,000?
Day 83 - What is a + b ?
Day 84 - What is the sum of the prime factors of the number represented by: 212 - 211 +210 - 29 + ...... +22 - 21?
Day 85 - What will row 50 look like? Can you generalize the result for any row of the triangle?
Day 85 - What will row 50 look like? Can you generalize the result for any row of the triangle?
Day 106 - Fill in the missing terms of this pattern: 102, 105, 111, 114, 120, 123, 129, ____, ____, ____, ____, ____, 201, 204, 210, 213, 219, .....
Day 107 - The positive integers are written in a triangular array as shown. In what row is the number 1000?
1
2 3
4 5 6
7 8 9 10 ...
Day 108 - What is the sum of the numbers in the 100th row of the triangular array in problem 107?
Day 109 - If this pattern continues, what is a possible next number in the sequence 1, 7, 25, 61, 121, ...?
Day 110 - The positive numbers are written in the pattern shown. Find the number in the 100th row and 100th column.
Day 107 - The positive integers are written in a triangular array as shown. In what row is the number 1000?
1
2 3
4 5 6
7 8 9 10 ...
Day 108 - What is the sum of the numbers in the 100th row of the triangular array in problem 107?
Day 109 - If this pattern continues, what is a possible next number in the sequence 1, 7, 25, 61, 121, ...?
Day 110 - The positive numbers are written in the pattern shown. Find the number in the 100th row and 100th column.
Day 131 - Fill in the blank with a number to complete the following pattern: ____, 1661, 1771, 1881, 1991, 2002, ...
Day 132 - If this lattice were continued, what number would be directly to the right of 98?
Day 132 - If this lattice were continued, what number would be directly to the right of 98?
Day 133 - What could the next four numbers in this progression be? 12, 1, 1, 1, 2, 1, 3, ___, ___, ___, ___
Day 134 - In what row and column is 1996?
Day 134 - In what row and column is 1996?
Day 135 - How many terms are in the following sequence? 10, 17, 24, 31, ..., 374
Often a systematic list or table can make the search for a solution to a problem much easier.
Example: In how many ways can you pile 25 marbles into 3 piles with an odd number in each pile? By organizing a list, all solutions can be found.
1, 1, 23
1, 3, 21
1, 5, 19
1, 7, 17
...
Day 16 - Take 25 marbles. Put them in 3 piles so an odd number is in each pile. How many ways can this be done?
Day 17 - A rectangle has an area of 120 sq. cm.. Its length and width are whole numbers. What are the possibilites for the two numbers? Which possibility gives the smallest perimeter?
Day 18 - The product of two whole numbers is 96 and their sum is less than 30. What are possibilities for the two numbers?
Day 19 - Jamie and Lynn each worked a different number of days, but each earned the same amount of money. Use the following clues to find how many days each worked: - Jamie earned $15 a day. - Lynn earned $10 a day. - Lynn worked 5 more days than Jamie.
Day 20 - Lonnie has a large supply of quarters, dimes, nickels, and pennies. In how many ways could she make change for 50 cents?
Day 41 - How many different four-digit numbers can be formed using the digits 1, 1, 9, and 9?
Day 42 - Two different prime numbers are selected at random from the first ten prime numbers. What is the probability that the sum of the two primes is 24?
Day 43 - Which is greater: $5.00 or the total value of all combinations of three coins you can make using only pennies, nickels, dimes, and quarters?
Day 44 - "Chicken Chunkettes" come in boxes of 6, 9, and 20. What is the largest number of chunkettes you can't buy?
Day 45 - In how many different ways is it possible to score 15 points in basketball?
Day 66 - A basketball player is on the line to shoot a 1 and 1 free throw. If the player's free throw average is .750, what is the probability that she will score exactly one basket?
Day 67 - Several sets of three different numbers whose sum is 15 can be chosen from 1, 2, 3, 4, 5, 6, 7, 8, 9. How many of these sets contain a 5?
Day 68 - How many positive integral factors does the number 720 have?
Day 69 - A football team boasts that all the numbers on the jerseys are prime numbers under 100. What is the largest number of players the team could have?
Day 70 - A three-digit number is selected at random from all three-digit numbers 100 - 999. What is the probability that the number is a perfect square?
Day 91 - How many four-digit numbers N have the following properties? (1) the sum of the digits of N is the same as the number obtained by deleting the last two digits of N. (2) the sum of the digits of N equals the product of the last two digits of N.
Day 92 - A social club contains 7 women and 4 men. The committee wants to select a committee of 3 members to represent it at the state convention. How many of the possible committees that could be chosen contain at least one man?
Day 93 - How many distinct isosceles triangles having sides of integral length and a perimeter of 113 are possible?
Day 94 - How many different secret code words can be made using three stars and two dashes in each word?
Day 95 - If you make $1000 every time the hands of a clock form a 90-degree angle, how much would you make in 24 hours?
Day 116 - How many decimal numerals are made up of the digits 1, 2, 3, 4, 5, each used at most once, and are also multiples of 8?
Day 117 - In how many ways can eight dollars be changed into dimes and/or quarters?
Day 118 - Radio stations use three or four letters for their call letters. The first letter must be a W or a K. How many different call- letter strings are possible if no letter may be repeated within a string?
Day 119 - A number is chosen at random from the following : .25, .5, .75, .8, 1, 2, 2.2, 3, 4, 9.7 What is the probability that its reciprocal is greater than one?
Day 120 - Myrtle has two white balls, two black balls, and two boxes. She may place the balls in the boxes in any way that she pleases. Her husband will then pick a box without looking into it, and with his eyes closed, pick out a ball. If he draws a white ball, the couple wins $500. How should Myrtle arrange the balls to maximize the probability of winning?
Day 141 - If each of these three operation signs, +, -, and x is used exactly once in the blanks in the expression 5 ___ 4 ___ 6 ___ 3 then how many different final values can you make?
Day 142 - How many three-digit numbers can be formed from the digits 0, 1, 2, 3, and 4 if no repetitions are allowed?
Day 143 - Using a deck of 52 cards, how many 5-card poker hands that contain 4 aces can you construct? Assume that no cards are wild.
Day 144 - The sum of three numbers is 98. The ratio of the first to the second is 2 to 3, and the ratio of the second to the third is 5 to 8. What is the second number?
Day 145 - A salesperson wants to rent a car for one day. Rental agency A charges $35 per day plus $0.20 per mile driven. Rental agency B charges $30 per day plus $0.25 per mile driven. Should she rent from Agency A or Agency B to get the best rate?
Example: In how many ways can you pile 25 marbles into 3 piles with an odd number in each pile? By organizing a list, all solutions can be found.
1, 1, 23
1, 3, 21
1, 5, 19
1, 7, 17
...
Day 16 - Take 25 marbles. Put them in 3 piles so an odd number is in each pile. How many ways can this be done?
Day 17 - A rectangle has an area of 120 sq. cm.. Its length and width are whole numbers. What are the possibilites for the two numbers? Which possibility gives the smallest perimeter?
Day 18 - The product of two whole numbers is 96 and their sum is less than 30. What are possibilities for the two numbers?
Day 19 - Jamie and Lynn each worked a different number of days, but each earned the same amount of money. Use the following clues to find how many days each worked: - Jamie earned $15 a day. - Lynn earned $10 a day. - Lynn worked 5 more days than Jamie.
Day 20 - Lonnie has a large supply of quarters, dimes, nickels, and pennies. In how many ways could she make change for 50 cents?
Day 41 - How many different four-digit numbers can be formed using the digits 1, 1, 9, and 9?
Day 42 - Two different prime numbers are selected at random from the first ten prime numbers. What is the probability that the sum of the two primes is 24?
Day 43 - Which is greater: $5.00 or the total value of all combinations of three coins you can make using only pennies, nickels, dimes, and quarters?
Day 44 - "Chicken Chunkettes" come in boxes of 6, 9, and 20. What is the largest number of chunkettes you can't buy?
Day 45 - In how many different ways is it possible to score 15 points in basketball?
Day 66 - A basketball player is on the line to shoot a 1 and 1 free throw. If the player's free throw average is .750, what is the probability that she will score exactly one basket?
Day 67 - Several sets of three different numbers whose sum is 15 can be chosen from 1, 2, 3, 4, 5, 6, 7, 8, 9. How many of these sets contain a 5?
Day 68 - How many positive integral factors does the number 720 have?
Day 69 - A football team boasts that all the numbers on the jerseys are prime numbers under 100. What is the largest number of players the team could have?
Day 70 - A three-digit number is selected at random from all three-digit numbers 100 - 999. What is the probability that the number is a perfect square?
Day 91 - How many four-digit numbers N have the following properties? (1) the sum of the digits of N is the same as the number obtained by deleting the last two digits of N. (2) the sum of the digits of N equals the product of the last two digits of N.
Day 92 - A social club contains 7 women and 4 men. The committee wants to select a committee of 3 members to represent it at the state convention. How many of the possible committees that could be chosen contain at least one man?
Day 93 - How many distinct isosceles triangles having sides of integral length and a perimeter of 113 are possible?
Day 94 - How many different secret code words can be made using three stars and two dashes in each word?
Day 95 - If you make $1000 every time the hands of a clock form a 90-degree angle, how much would you make in 24 hours?
Day 116 - How many decimal numerals are made up of the digits 1, 2, 3, 4, 5, each used at most once, and are also multiples of 8?
Day 117 - In how many ways can eight dollars be changed into dimes and/or quarters?
Day 118 - Radio stations use three or four letters for their call letters. The first letter must be a W or a K. How many different call- letter strings are possible if no letter may be repeated within a string?
Day 119 - A number is chosen at random from the following : .25, .5, .75, .8, 1, 2, 2.2, 3, 4, 9.7 What is the probability that its reciprocal is greater than one?
Day 120 - Myrtle has two white balls, two black balls, and two boxes. She may place the balls in the boxes in any way that she pleases. Her husband will then pick a box without looking into it, and with his eyes closed, pick out a ball. If he draws a white ball, the couple wins $500. How should Myrtle arrange the balls to maximize the probability of winning?
Day 141 - If each of these three operation signs, +, -, and x is used exactly once in the blanks in the expression 5 ___ 4 ___ 6 ___ 3 then how many different final values can you make?
Day 142 - How many three-digit numbers can be formed from the digits 0, 1, 2, 3, and 4 if no repetitions are allowed?
Day 143 - Using a deck of 52 cards, how many 5-card poker hands that contain 4 aces can you construct? Assume that no cards are wild.
Day 144 - The sum of three numbers is 98. The ratio of the first to the second is 2 to 3, and the ratio of the second to the third is 5 to 8. What is the second number?
Day 145 - A salesperson wants to rent a car for one day. Rental agency A charges $35 per day plus $0.20 per mile driven. Rental agency B charges $30 per day plus $0.25 per mile driven. Should she rent from Agency A or Agency B to get the best rate?
Sometimes making a drawing or model of the problem allows you to visualize how to find the solution.
Example: A 12" by 16" rectangular sheet is to have 2" squares cut out of each corner. Find the area.
Example: A 12" by 16" rectangular sheet is to have 2" squares cut out of each corner. Find the area.
Day 11 - Four squares are arranged so that each square touches at least one other square. Any two squares touch each other according to these rules: They can touch on the corners or they can touch entirely across one side but not partially across a side. What are the possible perimeters?
Day 12 - Billy, Ray, Gary, Clayton, and Pete just completed a 3000 meter race. Use these clues to help you determine the order in which they finished. - Floyd finished 10 seconds behind Gary. - Ray beat Pete by 20 seconds. - Billy finished 4 seconds behind Gary and 30 seconds ahead of Ray. - Clayton's finish was halfway between Floyd and Pete.
Day 13 - A 9 meter by 12 meter rectangular garden has a walk one meter wide all around it. What is the area of the walk?
Day 14 - One painter can paint a wall in ten minutes. Another painter can paint it in 6 minutes. About how long will it take both painters to do the job working together?
Day 15 - Julie can eat a medium-sized pepperoni pizza in 8 minutes. Her friend Samantha can eat the same pizza in 6 minutes. If both of them ate at their same rates, how long would it take them to eat one medium-sized pepperoni pizza together?
Day 37 - The number of diagonals of a regular polygon is subtracted from the number of sides of the polygon and the result is zero. What is the number of sides of this polygon?
Day 38 - What is the maximum number of points of intersection when two distinct circles and three distinct lines intersect each other?
Day 39 - A straight concrete sidewalk is to be 3 feet wide, 60 feet long, and 3 inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
Day 40 - If 120 seats are arranged in a row, what is the least number of seats that must be occupied so the next person seated must sit next to someone?
Day 61 - Two opposite sides of a square are increased by 25% while the other two are decreased by 40%. Find the percent of decrease in the area of the square.
Day 62 - Alice is 100 meters from Bill and Bill is 300 meters from Chelsea. They all move to the right at constant speeds. In 6 minutes, Alice overtakes Bill, and in another 6 minutes, Alice overtakes Chelsea. How many minutes did it take for Bill to overtake Chelsea?
Day 63 - A man is digging a hole and standing in it. He is 5 feet 10 inches tall. He says that he is one-fourth done and that when he is finished, the top of his head will be three times as far below the ground as it is now above the ground. How deep will the hole be when finished?
Day 64 - An officer on horseback starts at the back of a column of marching soldiers and rides to the front of the column, then turns around and rides back to the rear of the column. If the rider travels three times as fast as the column moves and the column is 100 meters long, how far does the column move by the time the officer completes the tour of inspection?
Day 65 - If the radius of a circle is increased by 20 percent, by what percent is the area increased?
Day 86 - True or false: An equation of the bisector of the angle formed by the x-axis and the line y = x is y = 1/2 x
Day 87 - 18 people numbered 1 - 18 are seated equally spaced around a circular table. What is the number of the person directly across from the person numbered 6?
Day 89 - A golfer hits a ball 250 yards but 2 degrees left of the intended line. How many yards from the intended spot did the ball land?
Day 90 - At a concert, Phillip, Sally, Gerald, and John sit in 4 seats in a row. Phillip must sit next to Sally but not next to Gerald. If Gerald will not sit next to John, who is sitting next to John?
Day 111 - 100 students were asked if they liked Country-Western or classical music. 5 students said they liked neither, 85 liked Country-Western, and 23 liked classical. How many students liked both?
Day 112 - A square is the only distinct monomino. Two squares is the only distinct domino. Triominoes, however, can be formed two ways (straight line or bent like an L). Draw all the distinct tetrominoes.
Day 114 - A man calculated that if he skis 10 km/hr, he will arrive at his cabin at 1:00 PM. If he skis at a rate of 15 km/hr, he will arrive at his cabin at 11:00 AM. How fast must he ski to arrive at his cabin at noon?
Day 115 - When the midpoints of the side of a square are joined, a new square is formed. If this process continues indefinitely, what is the sum of all the perimeters of the squares if the original square was 4 cm.?
Day 136 - What is the perimeter of a square with a diagonal of 36?
Day 137 - Determine the largest number of boxes of dimensions 2x2x3 that can be placed inside a box 3x4x5.
Day 138 - A hexagon is inscribed in a circle, which is inscribed in a square of side 10 cm. What is the length of each side of the hexagon?
Day 139 - A golf ball falls randomly into a circular green 10 meters in radius, with the cup at the center. What is the probability that the ball falls within 1 meter of the cup?
Day 140 - Three vertices of a parallelogram are ( 1, 1), ( 3, 5 ), and ( -1, 4 ). Find all possible ordered pairs that could be the coordinates of the fourth vertex.
Day 12 - Billy, Ray, Gary, Clayton, and Pete just completed a 3000 meter race. Use these clues to help you determine the order in which they finished. - Floyd finished 10 seconds behind Gary. - Ray beat Pete by 20 seconds. - Billy finished 4 seconds behind Gary and 30 seconds ahead of Ray. - Clayton's finish was halfway between Floyd and Pete.
Day 13 - A 9 meter by 12 meter rectangular garden has a walk one meter wide all around it. What is the area of the walk?
Day 14 - One painter can paint a wall in ten minutes. Another painter can paint it in 6 minutes. About how long will it take both painters to do the job working together?
Day 15 - Julie can eat a medium-sized pepperoni pizza in 8 minutes. Her friend Samantha can eat the same pizza in 6 minutes. If both of them ate at their same rates, how long would it take them to eat one medium-sized pepperoni pizza together?
Day 37 - The number of diagonals of a regular polygon is subtracted from the number of sides of the polygon and the result is zero. What is the number of sides of this polygon?
Day 38 - What is the maximum number of points of intersection when two distinct circles and three distinct lines intersect each other?
Day 39 - A straight concrete sidewalk is to be 3 feet wide, 60 feet long, and 3 inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
Day 40 - If 120 seats are arranged in a row, what is the least number of seats that must be occupied so the next person seated must sit next to someone?
Day 61 - Two opposite sides of a square are increased by 25% while the other two are decreased by 40%. Find the percent of decrease in the area of the square.
Day 62 - Alice is 100 meters from Bill and Bill is 300 meters from Chelsea. They all move to the right at constant speeds. In 6 minutes, Alice overtakes Bill, and in another 6 minutes, Alice overtakes Chelsea. How many minutes did it take for Bill to overtake Chelsea?
Day 63 - A man is digging a hole and standing in it. He is 5 feet 10 inches tall. He says that he is one-fourth done and that when he is finished, the top of his head will be three times as far below the ground as it is now above the ground. How deep will the hole be when finished?
Day 64 - An officer on horseback starts at the back of a column of marching soldiers and rides to the front of the column, then turns around and rides back to the rear of the column. If the rider travels three times as fast as the column moves and the column is 100 meters long, how far does the column move by the time the officer completes the tour of inspection?
Day 65 - If the radius of a circle is increased by 20 percent, by what percent is the area increased?
Day 86 - True or false: An equation of the bisector of the angle formed by the x-axis and the line y = x is y = 1/2 x
Day 87 - 18 people numbered 1 - 18 are seated equally spaced around a circular table. What is the number of the person directly across from the person numbered 6?
Day 89 - A golfer hits a ball 250 yards but 2 degrees left of the intended line. How many yards from the intended spot did the ball land?
Day 90 - At a concert, Phillip, Sally, Gerald, and John sit in 4 seats in a row. Phillip must sit next to Sally but not next to Gerald. If Gerald will not sit next to John, who is sitting next to John?
Day 111 - 100 students were asked if they liked Country-Western or classical music. 5 students said they liked neither, 85 liked Country-Western, and 23 liked classical. How many students liked both?
Day 112 - A square is the only distinct monomino. Two squares is the only distinct domino. Triominoes, however, can be formed two ways (straight line or bent like an L). Draw all the distinct tetrominoes.
Day 114 - A man calculated that if he skis 10 km/hr, he will arrive at his cabin at 1:00 PM. If he skis at a rate of 15 km/hr, he will arrive at his cabin at 11:00 AM. How fast must he ski to arrive at his cabin at noon?
Day 115 - When the midpoints of the side of a square are joined, a new square is formed. If this process continues indefinitely, what is the sum of all the perimeters of the squares if the original square was 4 cm.?
Day 136 - What is the perimeter of a square with a diagonal of 36?
Day 137 - Determine the largest number of boxes of dimensions 2x2x3 that can be placed inside a box 3x4x5.
Day 138 - A hexagon is inscribed in a circle, which is inscribed in a square of side 10 cm. What is the length of each side of the hexagon?
Day 139 - A golf ball falls randomly into a circular green 10 meters in radius, with the cup at the center. What is the probability that the ball falls within 1 meter of the cup?
Day 140 - Three vertices of a parallelogram are ( 1, 1), ( 3, 5 ), and ( -1, 4 ). Find all possible ordered pairs that could be the coordinates of the fourth vertex.
Example: Suppose we each have a bag of pennies. We take turns putting them on a rectangular table. No penny can hang over the edge and no pennies can overlap although they can touch. Whoever puts a penny in the last space available wins all the pennies. Would you want to play first or second? How would you play to win? What if you had a table only big enough to hold one penny?
That's pretty easy. You play first and win.
What about a two-penny sized table? Would you go first or second and where would you place your penny to guarantee a win?
What about a two-penny sized table? Would you go first or second and where would you place your penny to guarantee a win?
How about a three-penny sized table?
A four-penny sized table?
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(Hint: what would happen if you placed your penny in the middle?)Do you see that by simplifying the problem you can find a strategy to win every time?
Day 21 - Suppose we each have a bag of pennies. We take turns putting them on a rectangular table. No penny can hang over the edge and no pennies can overlap although they can touch. Whoever puts a penny in the last space available wins all the pennies. Would you want to play first or second? How would you play to win?
Day 22 - How many line segments can be drawn through each of 15 points? No three points are in a straight line.
Day 23 - Find the product: ( 1 - 1/2)(1 - 1/3)(1 - 1/4)......(1 - 1/98)(1 - 1/99)(1 - 1/100) = ?
Day 24 - Find the sum: 1/1x2 + 1/2x3 + 1/3x4 + ..... +1/98x99 + 1/99x100 = ?
Day 25 - Ten strangers attend a meeting. As introductions are made, each person shakes hands with all the others. How many handshakes occur?
Day 46 - If eighteen ounces of dough are used to make a sixteen-inch pizza, how many ounces are used for a twenty-inch pizza?
Day 48 - Jeremy travels from A to B at 2 minutes per mile and returns over the same route at 2 miles per minute. Find his average speed, in miles per hour, for the whole trip.
Day 49 - A ten meter pole and a forty meter pole are placed fifty meters apart on flat ground. Two nonsagging ropes join the top of one pole to the bottom of the other pole and vice versa. Calculate the height from the ground of the point of intersection of the ropes.
Day 50 - Four numbers are written in a row. The average of the first two numbers is 7, the average of the middle two numbers is 2.3, and the average of the last two numbers is 8.4. What is the average of the first number and the last number?
Day 71 - How many squares are in this picture?
Day 21 - Suppose we each have a bag of pennies. We take turns putting them on a rectangular table. No penny can hang over the edge and no pennies can overlap although they can touch. Whoever puts a penny in the last space available wins all the pennies. Would you want to play first or second? How would you play to win?
Day 22 - How many line segments can be drawn through each of 15 points? No three points are in a straight line.
Day 23 - Find the product: ( 1 - 1/2)(1 - 1/3)(1 - 1/4)......(1 - 1/98)(1 - 1/99)(1 - 1/100) = ?
Day 24 - Find the sum: 1/1x2 + 1/2x3 + 1/3x4 + ..... +1/98x99 + 1/99x100 = ?
Day 25 - Ten strangers attend a meeting. As introductions are made, each person shakes hands with all the others. How many handshakes occur?
Day 46 - If eighteen ounces of dough are used to make a sixteen-inch pizza, how many ounces are used for a twenty-inch pizza?
Day 48 - Jeremy travels from A to B at 2 minutes per mile and returns over the same route at 2 miles per minute. Find his average speed, in miles per hour, for the whole trip.
Day 49 - A ten meter pole and a forty meter pole are placed fifty meters apart on flat ground. Two nonsagging ropes join the top of one pole to the bottom of the other pole and vice versa. Calculate the height from the ground of the point of intersection of the ropes.
Day 50 - Four numbers are written in a row. The average of the first two numbers is 7, the average of the middle two numbers is 2.3, and the average of the last two numbers is 8.4. What is the average of the first number and the last number?
Day 71 - How many squares are in this picture?
©2016 Oregon Council of Teachers of Mathematics. All rights reserved.
Day 72 - Solve for x:
Day 73 - A certain king sent 30 men to plant trees. If they can set out 1000 trees in 9 days, how many days would it take for 36 men to set out 4400 trees?
Day 74 - A dog chasing a rabbit, which has a lead of 150 feet, jumps 9 feet for every time the rabbit jumps 7. In how many leaps does the dog overtake the rabbit?
Day 75 - Ten people are seated around a circular table. Each person shakes hands with everyone else except the people who sat on either side. How many handshakes take place?
Day 96 - A wagon train had 96 wagons , each carrying the same number of people. When 12 wagons broke down, each of the other wagons had to carry one more person. How many people were in each wagon originally?
Day 97 - What was a walker's average speed if she completed a trip around the square city block shown here?
Day 74 - A dog chasing a rabbit, which has a lead of 150 feet, jumps 9 feet for every time the rabbit jumps 7. In how many leaps does the dog overtake the rabbit?
Day 75 - Ten people are seated around a circular table. Each person shakes hands with everyone else except the people who sat on either side. How many handshakes take place?
Day 96 - A wagon train had 96 wagons , each carrying the same number of people. When 12 wagons broke down, each of the other wagons had to carry one more person. How many people were in each wagon originally?
Day 97 - What was a walker's average speed if she completed a trip around the square city block shown here?
Day 98 - In the 1973 Belmont Stakes, Secretariat covered 12 furlongs in 2 minutes, 24 seconds. What was his speed in miles per hour? (A furlong is 1/8 of a mile.)
Day 99 - A crowd watching a parade fills the sidewalks on both sides of the street for a distance of 2 miles. The sidewalks are 10 feet deep and an average person needs 4 square feet to stand on. A good estimate of the crowd is : (a) 25,000 people (b) 50,000 people (c) 100,000 people (d) 250,00 people (e) 500,000 people
Day 100 - A school has 1200 students. Each student takes 5 classes a day. Each teacher teaches 4 classes. Each class has 30 students and 1 teacher. How many teachers does the school have?
Day 121 - How many whole numbers between 100 and 400 contain the digit 2?
Day 122 - A cake has three circular tiers; each is 8 cm. high. The tiers have diameters of 60 cm., 48 cm., and 36 cm.. What is the surface area to be covered by frosting? There is no frosting between layers.
Day 123 - How many different routes can be traced from point A to point B if the movement can only be horizontal or vertical?
Day 99 - A crowd watching a parade fills the sidewalks on both sides of the street for a distance of 2 miles. The sidewalks are 10 feet deep and an average person needs 4 square feet to stand on. A good estimate of the crowd is : (a) 25,000 people (b) 50,000 people (c) 100,000 people (d) 250,00 people (e) 500,000 people
Day 100 - A school has 1200 students. Each student takes 5 classes a day. Each teacher teaches 4 classes. Each class has 30 students and 1 teacher. How many teachers does the school have?
Day 121 - How many whole numbers between 100 and 400 contain the digit 2?
Day 122 - A cake has three circular tiers; each is 8 cm. high. The tiers have diameters of 60 cm., 48 cm., and 36 cm.. What is the surface area to be covered by frosting? There is no frosting between layers.
Day 123 - How many different routes can be traced from point A to point B if the movement can only be horizontal or vertical?
Day 124 - In a 10 team conference where each team plays each other team at home, how many conference games are played in a season?
Day 125 - The density of wood of a pine tree is 35 pounds per cubic foot.The tree is 72 feet high and the diameter of the base is 3 feet. Ten percent of the weight of the tree is contained in the branches and foliage. If the trunk of the tree is a right circular cylinder, estimate the weight of the tree (above ground).
Day 147 - If 2 + 3 + 4 + ... + 1990 + 1991 + 1992 = 3N, then what is N?
Day 148 - 6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = which of these? (A) 6^6 (B) 6^7 (C) 36^6 (D) 6^36 (E) 36^36
Day 150 - If
Day 125 - The density of wood of a pine tree is 35 pounds per cubic foot.The tree is 72 feet high and the diameter of the base is 3 feet. Ten percent of the weight of the tree is contained in the branches and foliage. If the trunk of the tree is a right circular cylinder, estimate the weight of the tree (above ground).
Day 147 - If 2 + 3 + 4 + ... + 1990 + 1991 + 1992 = 3N, then what is N?
Day 148 - 6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = which of these? (A) 6^6 (B) 6^7 (C) 36^6 (D) 6^36 (E) 36^36
Day 150 - If
find (3*48)*9.
©2017 Oregon Council of Teachers of Mathematics. All rights reserved.