Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the game. When Annika and 3 of her friends play Buffalo Shuffle-o, each player is dealt 15 cards. Suppose 2 more friends join the next game. How many cards will be dealt to each player?
Show hint (soft nudge)
The total number of cards doesn't change. Find that total first.
Show hint (sharpest)
First find the total number of cards (it doesn't change), then share it among the new, larger group.
Show solution
Annika + 3 friends = 4 players, each dealt 15, so there are 4 × 15 = 60 cards.
With 2 more friends, there are now 4 + 2 = 6 players.
Four friends do yardwork for their neighbors over the weekend, earning $15, $20, $25, and $40, respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned $40 give to the others?
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Find the fair share first — what does each friend end up with?
Show hint (sharpest)
The $40 friend gives away whatever they have above the fair share.
Carrie has a rectangular garden that measures 6 feet by 8 feet. She plants the entire garden with strawberry plants. Carrie is able to plant 4 strawberry plants per square foot, and she harvests an average of 10 strawberries per plant. How many strawberries can she expect to harvest?
Show hint
Walk it through in units: square feet → plants → strawberries. Just multiply each step.
Ike and Mike go into a sandwich shop with a total of $30.00 to spend. Sandwiches cost $4.50 each and soft drinks cost $1.00 each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?
Show hint (soft nudge)
Spend on sandwiches first — what's the most they can buy without going past $30?
Show hint (sharpest)
6 sandwiches cost $27, leaving $3 for sodas.
Show solution
$30 ÷ $4.50 = 6 with $3 left over (a 7th sandwich would cost $31.50, too much).
Billy's basketball team scored the following points over the course of the first 11 games of the season: 42, 47, 53, 53, 58, 58, 58, 61, 64, 65, 73. If his team scores 40 in the 12th game, which of the following statistics will show an increase?
Show hint
40 is below the previous minimum (42), so it becomes the new low. Which statistic depends on the spread between min and max?
Show solution
Original range: 73 − 42 = 31. New range: 73 − 40 = 33 → increases.
Median (lowering by adding a small value) decreases or stays. Mean drops (40 is below current mean). Mode stays 58. Mid-range = (max+min)/2 decreases (min drops, max unchanged).
Harry and Terry are each told to calculate 8 − (2 + 5). Harry gets the correct answer. Terry ignores the parentheses and calculates 8 − 2 + 5. If Harry's answer is H and Terry's answer is T, what is the difference H − T?
Paul owes Paula 35 cents and has a pocket full of 5-cent coins, 10-cent coins, and 25-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
Show hint
Max coins → all 5-cent coins. Min coins → use the biggest coins possible.
Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. How many pages were in the book?
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Average × count gives the total for each three-day block. Then add the last day.
The first AMC 8 was given in 1985 and it has been given annually since that time. Samantha turned 12 years old the year that she took the seventh AMC 8. In what year was Samantha born?
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The n-th AMC 8 was given in 1985 + (n − 1). Subtract her age to get her birth year.
Hammie is in the 6th grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
Show hint
Order the 5 weights to find the median. Mean = total / 5. The single outlier (106) pulls the mean way up.
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, 30 = 6 × 5. What is the missing number in the top row?
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was 6:57 AM, and the sunset as 8:15 PM. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set?
Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
Show hint (soft nudge)
Total points needed = 4 × 95 = 380. Subtract the first two scores to find what's left for tests 3 + 4.
Show hint (sharpest)
To minimize the 3rd test, maximize the 4th (cap = 100).
Here is a list of the numbers of fish that Tyler caught in nine outings last summer: 2, 0, 1, 3, 0, 3, 3, 1, 2. Which statement about the mean, median, and mode is true?
Show hint
Sort the list. Mode = most common value, median = middle (5th of 9), mean = sum/9.
The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?
Show hint
Add the daily differences (Sasha − Asha) and divide by 5.
At Euclid Middle School the mathematics teachers are Miss Germain, Mr. Newton, and Mrs. Young. There are 11 students in Mrs. Germain's class, 8 students in Mr. Newton's class, and 9 students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid Middle School are taking the contest?
Alice needs to replace a light bulb located 10 centimeters below the ceiling in her kitchen. The ceiling is 2.4 meters above the floor. Alice is 1.5 meters tall and can reach 46 centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?
The average age of the 6 people in Room A is 40. The average age of the 4 people in Room B is 25. If the two groups are combined, what is the average age of all the people?
Show hint
Combined average = total ages / total people. Each room's total = avg × count.
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of 10 hours per week helping around the house for 6 weeks. For the first 5 weeks she helps around the house for 8, 11, 7, 12 and 10 hours. How many hours must she work for the final week to earn the tickets?
Show hint
Target total = 6 · 10 = 60. Subtract what she's already done.
650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?
Chandler wants to buy a 500 dollar mountain bike. For his birthday, his grandparents send him 50 dollars, his aunt sends him 35 dollars and his cousin gives him 15 dollars. He earns 16 dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike?
Show hint
Subtract birthday total from 500, then divide by 16.
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? (right = +1, wrong or N/A = +0)
Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer?
Show hint
First recover the original number, then divide by 2.
An athlete's target heart rate, in beats per minute, is 80% of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from 220. To the nearest whole number, what is the target heart rate of an athlete who is 26 years old?
Handy Aaron helped a neighbor 114 hours on Monday, 50 minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $3 per hour. How much did he earn for the week?
Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from 10, doubles his answer, and then adds 2. Thuy doubles 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from 10, adds 2 to his number, and then doubles the result. Who gets the largest final answer?
The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array). The first 8 numbers go in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be
Show hint (soft nudge)
The top row is 1–8 and the bottom row is 57–64.
Show hint (sharpest)
The corners are the first and last entry of those two rows.
Show solution
The corners are 1 and 8 (top row) and 57 and 64 (bottom row).
Two integers are inserted into the list 3, 3, 8, 11, 28 to double its range. The mode and median remain unchanged. What is the maximum possible sum of two additional numbers?
Show hint (soft nudge)
Original range = 28 − 3 = 25. New range = 50. To maximize the sum, push the new max as high as possible.
Show hint (sharpest)
Keep min = 3 ⇒ new max = 3 + 50 = 53. That's one insert. Now find the largest second insert that keeps median = 8 and mode = 3.
Show solution
Range doubles from 25 to 50. To maximize the sum, leave the min at 3 and stretch the max: one insert = 3 + 50 = 53.
With 7 numbers, the median is the 4th. Sorted so far: 3, 3, 8, 11, 28, 53 (6 values). The 2nd insert x must keep median = 8.
If x > 8, the 4th value shifts off 8 (and choosing x = 8 ties the mode with two 8's). So x ≤ 7.
Maximum x = 7 (mode stays uniquely 3). Sum = 53 + 7 = 60.
Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating 3 inches of pasta from the middle of one piece. In the end, he has 10 pieces of pasta whose total length is 17 inches. How long, in inches, was the piece of pasta he started with?
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Each bite turns one piece into two. How many bites does it take to end with 10 pieces?
Four numbers are written in a row. The average of the first two is 21, the average of the middle two is 26, and the average of the last two is 30. What is the average of the first and last of the numbers?
Show hint (soft nudge)
Convert each average into a sum (multiply by 2). You don't need the four numbers individually — just a + d.
Show hint (sharpest)
(a + b) + (c + d) = (a+b+c+d), and you can subtract (b+c) to leave a+d.
Show solution
a + b = 42, b + c = 52, c + d = 60.
(a+b) + (c+d) − (b+c) = a + d = 42 + 60 − 52 = 50.
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?
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Compare each day to the 85-minute target instead of adding everything up.
Show hint (sharpest)
The first 5 days fall short of 85 and the next 3 go over — track the running shortfall.
Show solution
Against an 85-minute target, each of the 5 days at 75 min is 10 short (−50 total), and each of the 3 days at 90 min is 5 over (+15 total).
That leaves a shortfall of 50 − 15 = 35 minutes, so day 9 must be 85 + 35 = 120 minutes = 2 hours.
E
2 hours.
Another way: totals
Skated so far: 5·75 + 3·90 = 645 min. Needed for the average: 9·85 = 765 min.
A set of five positive integers has mean 5, median 5, and 8 as its only mode. What is the difference between the largest and smallest integers in the set?
Show hint (soft nudge)
The five numbers sum to 5 × 5 = 25, and the mode 8 means two of them are 8.
Show hint (sharpest)
The median 5 is the middle value; the two smallest must be distinct and add to what's left.
Show solution
The numbers total 25; two 8s (the only mode) account for 16, and the median forces the middle value 5, leaving 4 for the two smallest.
Distinct positive integers adding to 4 are 1 and 3, giving {1, 3, 5, 8, 8} and a difference of 8 − 1 = 7.
If each of the three operation signs +, −, × is used exactly once in one of the blanks in the expression 5 __ 4 __ 6 __ 3, then the value of the result could equal
Show hint (soft nudge)
Remember multiplication happens before addition and subtraction.
Show hint (sharpest)
Try placing × between 6 and 3.
Show solution
Take 5 − 4 + 6 × 3: the multiplication gives 18 first.
One proposal for new postage rates for a letter was 30 cents for the first ounce and 22 cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing 4.5 ounces was
Show hint (soft nudge)
After the first ounce, 3.5 ounces remain — each fraction counts as a whole charge.
Show hint (sharpest)
Round 3.5 up to 4 additional charges.
Show solution
After the first ounce (30¢), 3.5 ounces remain, charged as 4 additional ounces.
The annual incomes of 1,000 families range from 8200 dollars to 98,000 dollars. In error, the largest income was entered on the computer as 980,000 dollars. The difference between the mean of the incorrect data and the mean of the actual data is
Show hint (soft nudge)
Only one entry changed, so the totals differ by that one error.
Show hint (sharpest)
Divide the size of the error by the number of families.
Show solution
The wrong entry overstates the total by 980,000 − 98,000 = 882,000.
Over 1,000 families, the mean is off by 882,000 ÷ 1000 = $882.
In January 1980 the Mauna Loa Observatory recorded carbon dioxide CO2 levels of 338 ppm (parts per million). Over the years the average CO2 reading has increased by about 1.515 ppm each year. What is the expected CO2 level in ppm in January 2030? Round your answer to the nearest integer.
Show hint
Multiply the per-year rate by the number of years, then add to the starting value.
Shauna takes five tests, each worth a maximum of 100 points. Her scores on the first three tests are 76, 94, and 87. In order to average 81 for all five tests, what is the lowest score she could earn on one of the other two tests?
Show hint (soft nudge)
To minimize one of the two remaining scores, max out the other (= 100). Then the rest is forced.
Show hint (sharpest)
Total needed: 5 × 81 = 405. First three sum to 257. Remaining two must sum to 148, so the minimum is 148 − 100.
Show solution
Total required: 5 × 81 = 405. First three tests: 76 + 94 + 87 = 257.
Remaining two tests must total 405 − 257 = 148.
To minimize one, set the other to 100: minimum = 148 − 100 = 48.
Blake and Jenny each took four 100-point tests. Blake averaged 78 on the four tests. Compared with Blake, Jenny scored 10 points higher on the first test, 10 points lower on the second, and 20 points higher on each of the third and fourth. By how much does Jenny's average exceed Blake's on these four tests?
Show hint
You never need Blake's actual average — just track how far ahead or behind Jenny is on each test.
Show solution
Jenny's differences from Blake are +10, −10, +20, +20, which total +40.
Art, Roger, Paul, and Trisha bake cookies that are all the same thickness, in the shapes shown below (dimensions in inches). Each friend uses the same amount of dough, and Art's batch makes exactly 12 cookies.
How many cookies will be in one batch of Trisha's cookies?
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A batch is the dough that makes 12 of Art's cookies. How does Trisha's cookie size compare to Art's?
Show solution
A batch is the dough for 12 of Art's 12 in² cookies, i.e. 144 in².
Trisha's cookies are ½(3)(4) = 6 in², exactly half of Art's, so she makes twice as many: 24.
Juan organizes the stamps in his collection by country and by the decade in which they were issued. He paid these prices at the stamp shop: Brazil and France, 6¢ each; Peru, 4¢ each; and Spain, 5¢ each. (Brazil and Peru are South American countries; France and Spain are European.) The table shows how many stamps he has from each country and decade.
How many of his European stamps were issued in the 1980s?
Number of Stamps by Decade
Country
'50s
'60s
'70s
'80s
Brazil
4
7
12
8
France
8
4
12
15
Peru
6
4
6
10
Spain
3
9
13
9
Show hint (soft nudge)
France and Spain are the European countries.
Show hint (sharpest)
Add their two entries in the 1980s column.
Show solution
The European countries are France and Spain.
In the 1980s they have 15 and 9 stamps: 15 + 9 = 24.
Juan organizes the stamps in his collection by country and by the decade in which they were issued. He paid these prices at the stamp shop: Brazil and France, 6¢ each; Peru, 4¢ each; and Spain, 5¢ each. (Brazil and Peru are South American countries; France and Spain are European.) The table shows how many stamps he has from each country and decade.
His South American stamps issued before the 1970s cost him how much?
Number of Stamps by Decade
Country
'50s
'60s
'70s
'80s
Brazil
4
7
12
8
France
8
4
12
15
Peru
6
4
6
10
Spain
3
9
13
9
Show hint (soft nudge)
South American means Brazil and Peru; “before the 1970s” means the '50s and '60s columns.
Show hint (sharpest)
Multiply each country's stamp count by its price, then add.
Show solution
Brazil and Peru are South American; before the 1970s covers the '50s and '60s.
Juan organizes the stamps in his collection by country and by the decade in which they were issued. He paid these prices at the stamp shop: Brazil and France, 6¢ each; Peru, 4¢ each; and Spain, 5¢ each. (Brazil and Peru are South American countries; France and Spain are European.) The table shows how many stamps he has from each country and decade.
The average price of his 1970s stamps is closest to which value?
Number of Stamps by Decade
Country
'50s
'60s
'70s
'80s
Brazil
4
7
12
8
France
8
4
12
15
Peru
6
4
6
10
Spain
3
9
13
9
Show hint (soft nudge)
Average price = total cost of the 1970s stamps ÷ how many there are.
Show hint (sharpest)
Use the whole 1970s column across all four countries.
Show solution
1970s stamps: Brazil 12 and France 12 at 6¢, Peru 6 at 4¢, Spain 13 at 5¢.
Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees?
Show hint (soft nudge)
Count the gaps between trees, not the trees themselves.
Each row is 100 ft long — find the cheapest way to fill one row.
Show hint (sharpest)
Staggering the joints forces the alternate rows to use a 1-ft block at each end.
Show solution
A row of all 2-ft blocks needs 100 ÷ 2 = 50 blocks. To stagger the joints, the in-between rows use 49 two-ft blocks plus a 1-ft block on each end = 51 blocks.
Four rows of 50 and three rows of 51: 200 + 153 = 353 blocks.
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults?
Show hint (soft nudge)
Work with total ages, not averages: everyone's ages add to 40 × 17.
Show hint (sharpest)
Subtract the girls' and boys' totals to leave the adults' total.
Show solution
All 40 ages total 40 × 17 = 680. Girls total 20 × 15 = 300 and boys total 15 × 16 = 240.
Adults total 680 − 300 − 240 = 140, so their average is 140 ÷ 5 = 28.
Cookies for a Crowd. At a school, 108 students eat an average of 2 cookies apiece. The recipe makes a pan of 15 cookies and uses 2 eggs per pan, and only full recipes are made. Walter buys eggs by the half-dozen. How many half-dozens should he buy to make enough cookies?
Show hint (soft nudge)
First find how many full pans of 15 cookies cover all the cookies needed.
Show hint (sharpest)
Then count the eggs and split them into half-dozens (6 each).
Show solution
The students eat 108 × 2 = 216 cookies, needing 216 ÷ 15 = 14.4 → 15 full pans.
Cookies for a Crowd. The recipe makes a pan of 15 cookies using 3 tablespoons of butter, and only full recipes are made. Walter and Gretel must supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter are needed?
Show hint (soft nudge)
Find the number of full pans for 216 cookies, then the butter they use.
Show hint (sharpest)
Convert tablespoons to sticks (8 per stick), rounding up.
Show solution
216 cookies need 216 ÷ 15 = 14.4 → 15 pans, using 15 × 3 = 45 tablespoons of butter.
At 8 tablespoons per stick, that's 45 ÷ 8 = 5.6 → 6 sticks.
Walter catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours of additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?
Show hint (soft nudge)
Find the whole stretch from 7:30 a.m. to 4:00 p.m. in minutes.
Show hint (sharpest)
Subtract all the time accounted for at school.
Show solution
From 7:30 a.m. to 4:00 p.m. is 8.5 hours = 510 minutes; school uses 6·50 + 30 + 2·60 = 450 minutes.
The bus rides take the rest: 510 − 450 = 60 minutes.
A teacher tells the class: "Think of a number, add 1 to it, and double the result. Give the answer to your partner. Partner, subtract 1 from the number you are given and double the result to get your answer." Ben thinks of 6 and gives his answer to Sue. What should Sue's answer be?
Show hint (soft nudge)
Compute Ben's answer first, then feed it into Sue's steps.
Each day Maria must work 8 hours. This does not include the 45 minutes she takes for lunch. If she begins working at 7:25 A.M. and takes her lunch break at noon, then her working day will end at
Show hint (soft nudge)
Find how much she works before lunch, then how much is left.
Show hint (sharpest)
Don't forget the 45-minute lunch pushes the afternoon back.
Show solution
From 7:25 to noon she works 4 h 35 min, leaving 8 h − 4 h 35 min = 3 h 25 min.
Lunch ends at 12:45, and 3 h 25 min later is 4:10 P.M.
The mean of a set of five different positive integers is 15. The median is 18. The maximum possible value of the largest of these five integers is
Show hint (soft nudge)
Mean 15 means the five numbers add to 75.
Show hint (sharpest)
To push one number as high as possible, make the other four as small as the rules allow.
Show solution
Five numbers averaging 15 sum to 5 × 15 = 75.
The median (3rd value) is 18, so the two below it are the smallest distinct positives 1 and 2, and the 4th value is the smallest integer above 18, namely 19.
There is a list of seven numbers. The average of the first four numbers is 5, and the average of the last four numbers is 8. If the average of all seven numbers is 647, then the number common to both sets of four numbers is
Show hint (soft nudge)
Add the two group-sums together — every number is counted once, except the shared one, counted twice.
Show hint (sharpest)
So (sum of the two fours) minus (sum of all seven) leaves exactly the common number.
Show solution
The first four total 4 · 5 = 20 and the last four total 4 · 8 = 32. Adding gives 52, which counts every number once except the shared middle one, counted twice.
All seven total 7 · 6⁴⁄₇ = 46, counting each number once.
Subtracting strips one copy of everything, leaving the doubled number: 52 − 46 = 6.
