A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?
Show hint (soft nudge)
The fence length stays the same, so find the square's side from that perimeter.
Show hint (sharpest)
Then compare the two areas.
Show solution
The fence is 2(50 + 10) = 120 ft, so the square has side 120 ÷ 4 = 30 ft and area 30² = 900.
The original area was 50 × 10 = 500, so the gain is 900 − 500 = 400 square feet.
Bo, Coe, Flo, Jo, and Moe have different amounts of money. Neither Jo nor Bo has as much money as Flo. Both Bo and Coe have more than Moe. Jo has more than Moe, but less than Bo. Who has the least amount of money?
Show hint (soft nudge)
You don't need the full ranking — just find who everyone beats.
Show hint (sharpest)
Every clue that names Moe places someone above him.
Show solution
Bo, Coe, and Jo are each said to have more than Moe, and Flo has more than Bo, so Flo beats Moe too.
The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?
Show hint (soft nudge)
The two exits are 160 − 40 = 120 mileposts apart.
Show hint (sharpest)
Go three-fourths of that distance past milepost 40.
Show solution
From milepost 40 to 160 is 120 miles, and three-fourths of 120 is 90.
A complete cycle of a traffic light takes 60 seconds. During each cycle the light is green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly chosen time, what is the probability that the light will NOT be green?
Show hint (soft nudge)
"Not green" just means yellow or red.
Show hint (sharpest)
Compare that time to the full 60-second cycle.
Show solution
Not green means yellow or red: 5 + 30 = 35 seconds out of 60.
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is 114. To the nearest whole percent, what percent of its games did the team lose?
Show hint (soft nudge)
Turn the ratio into parts: 11 won and 4 lost make 15 games in all.
Show hint (sharpest)
The lost fraction is 4 out of 15.
Show solution
Treat the ratio as 11 wins and 4 losses, so 15 games total.
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.
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set {C, H, L, P, R}, the second from {A, I, O}, and the third from {D, M, N, T}. When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set, or one letter may be added to one set and one to another. What is the largest possible number of additional license plates that can be made by adding two letters?
Show hint (soft nudge)
The number of plates is the product of the three set sizes — right now 5 × 3 × 4 = 60.
Show hint (sharpest)
Growing the smallest factors multiplies the count the most; try a couple of placements.
Show solution
Now there are 5 × 3 × 4 = 60 plates; two new letters change one or two of the factors.
The best is to enlarge the small factors: 5 × 5 × 4 (both into the size-3 set) or 5 × 4 × 5 each give 100 plates.
Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered 70% of the arithmetic, 40% of the algebra, and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% of the problems right. How many more problems would she have needed to answer correctly to earn a 60% passing grade?
Show hint (soft nudge)
Count how many she actually got right in each subject.
Show hint (sharpest)
Compare that total to 60% of all 75 problems.
Show solution
She got 7 of the arithmetic, 12 of the algebra, and 21 of the geometry right: 7 + 12 + 21 = 40 correct.
A 60% grade needs 0.6 × 75 = 45 correct, so she was short by 5.
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, and only full recipes are made. Normally 108 students each eat 2 cookies, but a concert cuts attendance by 25%. How many recipes should Walter and Gretel make for the smaller party?
Show hint (soft nudge)
A 25% drop leaves three-fourths of the 108 students.
Show hint (sharpest)
Find their cookies, then round up to whole pans of 15.
Show solution
Three-fourths of 108 is 81 students, eating 81 × 2 = 162 cookies.
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.
Each marked angle has a supplement along its line — start by finding those.
Show hint (sharpest)
Chase through the small triangles, using vertical angles where lines cross, until you reach A.
Show solution
The 100° and 110° marks have supplements 80° and 70° along their lines.
In the triangle holding the 40° tip and that 70°, the third angle is 180° − 70° − 40° = 70°, and its vertical angle at the crossing near A is also 70°.
In a far-off land three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth?
Show hint (soft nudge)
Convert bread into rice first, so everything is measured in rice.
Show hint (sharpest)
Then 3 fish equals that many bags — divide by 3.
Show solution
Since 1 loaf = 4 bags of rice, 2 loaves = 8 bags, and 3 fish trade for those 2 loaves.
So 3 fish = 8 bags, making one fish 8 ÷ 3 = 2⅔ bags of rice.
