Brynn's savings decreased by 20% in July, then increased by 50% of the new amount in August. Brynn's savings are now what percent of the original amount?
Show answer
Answer: E — 120%.
Show hints
Hint 1 of 2
Don't pick a starting amount — that's extra work. Each percent change is something you can just multiply by.
Still stuck? Show hint 2 →
Hint 2 of 2
Each percent change is just a multiplier — you don't even need a starting amount. Multiply the two.
Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive?
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Answer: C — 15%.
Show hints
Hint 1 of 2
Split 34 into 5 equal shares — what fraction of the whole pitcher is each share?
Still stuck? Show hint 2 →
Hint 2 of 2
Each cup gets 34 ÷ 5 = 320 of the pitcher. Now turn that into a percent.
A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk.)
Show answer
Answer: D — $12.
Show hint
Hint 1
First scale half-pound to full-pound; then undo the 50% discount.
Show solution
Approach: double for full pound, double again to undo 50% off
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?
Show answer
Answer: B — 30%.
Show hint
Hint 1
Compute the dollar saved on the 2nd and 3rd pairs (the 1st is full price). Then divide by $150.
Show solution
Approach: dollars saved / regular price
2nd pair saves 40% of $50 = $20. 3rd pair saves half of $50 = $25.
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?
Show answer
Answer: C — 1/8.
Show hint
Hint 1
Peter ate 1 full slice + half a slice = 1.5 slices out of 12.
A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price?
Show answer
Answer: D — 60%.
Show hint
Hint 1
Multiply the surviving fractions: 50% × 80% = 40% of original. Customer saves 100% − 40%.
Show solution
Approach: multiply remaining fractions
After 50% off: 0.5 of original. After 20% off that: 0.5 × 0.8 = 0.4 of original.
Ryan got 80% of the problems correct on a 25-problem test, 90% on a 40-problem test, and 70% on a 10-problem test. What percent of all the problems did Ryan answer correctly?
Show answer
Answer: D — 84%.
Show hint
Hint 1
Count correct on each test, then divide by the total number of problems.
In 2005 Tycoon Tammy invested 100 dollars for two years. During the first year her investment suffered a 15% loss, but during the second year the remaining investment showed a 20% gain. Over the two-year period, what was the change in Tammy's investment?
The average cost of a long-distance call in the USA in 1985 was 41 cents per minute, and the average cost of a long-distance call in the USA in 2005 was 7 cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call.
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?
Show answer
Answer: C — 40%.
Show hint
Hint 1
Yellow: 0.30 · 30 = 9 L. After adding 5: 14 L out of 35 L.
Initially, a spinner points west. Chenille moves it clockwise 214 revolutions and then counterclockwise 334 revolutions. In what direction does the spinner point after the two moves?
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Answer: B — East.
Show hint
Hint 1
Net rotation = clockwise − counterclockwise. Reduce mod 1 revolution.
Show solution
Approach: subtract revolutions, reduce mod 1
Net counterclockwise: 3.75 − 2.25 = 1.5 revolutions counterclockwise.
1.5 mod 1 = 0.5 ⇒ spinner turns half-revolution from west ⇒ east.
The table shows some of the results of a survey by radio station KACL. What percentage of the males surveyed listen to the station? (Total surveyed: 200. Females: 96. Females who listen: 58. Males who don't listen: 26. Total listeners: 136. Total non-listeners: 64.)
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Answer: E — 75%.
Show hint
Hint 1
Total males = 200 − 96 = 104. Of those, 26 don't listen.
Antonette gets 70% on a 10-problem test, 80% on a 20-problem test and 90% on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score?
Karl bought five folders from Pay-A-Lot at a cost of $2.50 each. Pay-A-Lot had a 20%-off sale the following day. How much could Karl have saved on the purchase by waiting a day?
The sales tax rate in Bergville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?
Show answer
Answer: C — $0.
Show hint
Hint 1
Multiplication is commutative: order of factors doesn't change the product.
After Sally takes 20 shots, she has made 55% of her shots. After she takes 5 more shots, she raises her percentage to 56%. How many of the last 5 shots did she make?
Show answer
Answer: C — 3.
Show hint
Hint 1
Made so far: 0.55 · 20 = 11. Target made: 0.56 · 25 = 14. Difference is how many she made in the last 5.
Two 600 mL pitchers contain orange juice. One pitcher is 1/3 full and the other pitcher is 2/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
Show answer
Answer: C — 11/30.
Show hint
Hint 1
Total OJ: 600(1/3) + 600(2/5) = 200 + 240 = 440. Total volume: 1200.
Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?
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Answer: C — 4/15.
Show hints
Hint 1 of 2
To minimize the overlap of "red AND high-top", let the OTHER kind (white) soak up as many high-tops as it can.
Still stuck? Show hint 2 →
Hint 2 of 2
9 red, 6 white. 10 high-top, 5 low-top. Make all 6 whites high-top; only 4 high-top spots remain — those must be red.
Show solution
Approach: push white pairs into high-top to crowd out red
For each weight, find the lowest price dot in that column. Then compute price ÷ weight.
Still stuck? Show hint 2 →
Hint 2 of 2
Lowest-price-per-ounce will favor a weight where the cheapest available pepper drops well below the dollar-per-ounce line.
Show solution
Approach: lowest dot in each column, then divide
Lowest price at each weight (reading off the scatter): 1 oz ≈ $1.25 (rate ≈ 1.25), 2 oz ≈ $2 (1.00), 3 oz ≈ $2.5 (≈ 0.83), 4 oz ≈ $3.9 (≈ 0.97), 5 oz ≈ $4.5 (≈ 0.90).
The 3-ounce option has the lowest rate (~$0.83/oz). Answer: 3 ounces.
Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a 60% chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
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Answer: B — 9 purple socks added.
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Hint 1 of 2
If s purple socks are added, what fraction of the new drawer is purple? Set that equal to 0.6.
Still stuck? Show hint 2 →
Hint 2 of 2
(18 + s) / (36 + s) = 0.6.
Show solution
Approach: non-purple count is fixed at 18 β and that's 40% of the new total
The green + orange socks (6 + 12 = 18) don't change. After adding purple, 60% of the drawer is purple, so the non-purple 18 socks make up the remaining 40%.
New total = 18 / 0.4 = 45 socks. Started with 36, so Jamal added 45 − 36 = 9 purple socks.
Another way — set up the new probability and solve:
If s purple are added, (18 + s) / (36 + s) = 0.6.
Cross-multiply: 18 + s = 21.6 + 0.6s, so 0.4s = 3.6 and s = 9.
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80% of the problems she solved alone, but overall 88% of her answers were correct. Zoe had correct answers to 90% of the problems she solved alone. What was Zoe's overall percentage of correct answers?
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Answer: C — 93%.
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Hint 1 of 2
Pretend there are 100 problems — 50 alone, 50 together. The "together" half is the same for both girls. Find that.
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Hint 2 of 2
Chloe: 80% of 50 = 40 alone; 88% total = 88 correct; so together = 48 of 50.
Show solution
Approach: use 100 problems, isolate the joint half
Jefferson Middle School has the same number of boys and girls. 34 of the girls and 23 of the boys went on a field trip. What fraction of the students on the field trip were girls?
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Answer: B — 9/17.
Show hint
Hint 1
Common denominator: 3/4 = 9/12 and 2/3 = 8/12. So per 12 girls there are 9 on the trip; per 12 boys, 8.
Show solution
Approach: common denominator gives the ratio
Same group sizes ⇒ girls : boys = 1 : 1. Their trip fractions: 3/4 vs 2/3 → 9/12 vs 8/12.
A merchant offers a large group of items at 30% off. Later, the merchant takes 20% off these sale prices and claims that the final price of these items is 50% off the original price. The total discount is
Show answer
Answer: B — 44%.
Show hints
Hint 1 of 2
Discounts don't simply add β track the fraction of the price you still pay.
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Hint 2 of 2
After 30% off you pay 0.7 of the price; the next 20% off pays 0.8 of that.
Show solution
Approach: multiply the fractions of price still paid
After 30% off you pay 0.70 of the original; taking another 20% off pays 0.80 of that.
So you pay 0.70 Γ 0.80 = 0.56 of the original β a 44% total discount, not 50%.
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
Show answer
Answer: D — 50 degrees.
Show hints
Hint 1 of 2
Find how many students like cherry pie first.
Still stuck? Show hint 2 →
Hint 2 of 2
Each student is worth 360Β° Γ· 36 = 10Β° of the circle.
Show solution
Approach: students β fraction of 360Β°
Remaining students: 36 β 12 β 8 β 6 = 10, and half prefer cherry, so 5 students.
Each student is 360Β° Γ· 36 = 10Β°, so cherry pie gets 5 Γ 10Β° = 50Β°.
For the game show Who Wants To Be a Millionaire?, the dollar values of each question are shown in the following table (where K = 1000). Between which two questions is the percent increase of the value the smallest?
Question values (K = 1000)
Question
1
2
3
4
5
6
7
8
Value
100
200
300
500
1K
2K
4K
8K
Question
9
10
11
12
13
14
15
Value
16K
32K
64K
125K
250K
500K
1000K
Show answer
Answer: B — From 2 to 3.
Show hints
Hint 1 of 2
Most steps simply double the value β that's a 100% increase, so ignore them.
Still stuck? Show hint 2 →
Hint 2 of 2
Only 2β3, 3β4, and 11β12 are not doublings; compare those.
Show solution
Approach: ignore the doublings, compare the exceptions
Almost every step doubles (a 100% jump), so the smallest increase is among the non-doublings: 2β3, 3β4, and 11β12.
2β3 is 200β300 = +50%; 3β4 is 300β500 β +67%; 11β12 is 64Kβ125K β +95%.
Three bags of jelly beans contain 26, 28, and 30 beans. The fractions of yellow beans in the bags are 50%, 25%, and 20%, respectively. All three bags are poured into one bowl. Which of the following is closest to the percent of yellow beans in the bowl?
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Answer: A — About 31%.
Show hints
Hint 1 of 2
Count the yellow beans in each bag, then total them.
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Hint 2 of 2
Divide the yellow total by the grand total of beans.
Show solution
Approach: yellow total over grand total
Yellow beans: 13 + 7 + 6 = 26, out of 26 + 28 + 30 = 84 beans.
Penni buys $100 of stock in each of three companies: AA, BB, and CC. After one year AA is up 20%, BB is down 25%, and CC is unchanged. In the second year AA drops 20% from its new value, BB rises 25% from its new value, and CC is unchanged. If A, B, C are the final values, which ordering is correct?
Show answer
Answer: E — B < A < C.
Show hints
Hint 1 of 2
Track each $100 through both years.
Still stuck? Show hint 2 →
Hint 2 of 2
A 20% rise followed by a 20% fall does not return to the start.
Last week small boxes of facial tissue were priced at 4 boxes for $5. This week they are on sale at 5 boxes for $4. The percent decrease in the price per box during the sale was closest to
Show answer
Answer: B — About 35%.
Show hints
Hint 1 of 2
Find the price of one box before and during the sale.
Still stuck? Show hint 2 →
Hint 2 of 2
Percent decrease compares the drop to the original price.
Show solution
Approach: per-box price before vs. during the sale
A box cost $5 Γ· 4 = $1.25 before and $4 Γ· 5 = $0.80 on sale.
The decrease is $0.45 Γ· $1.25 = 36%, closest to 35%.
A jacket and a shirt originally sold for 80 dollars and 40 dollars, respectively. During a sale Chris bought the 80-dollar jacket at a 40% discount and the 40-dollar shirt at a 55% discount. The total amount saved was what percent of the total of the original prices?
Show answer
Answer: A — 45%.
Show hints
Hint 1 of 2
Find the dollars saved on each item.
Still stuck? Show hint 2 →
Hint 2 of 2
Compare the total saved to the total original price of $120.
Show solution
Approach: total saved over total original
Savings: 40% of $80 = $32, and 55% of $40 = $22, for $54 saved.
A shopper buys a 100-dollar coat on sale for 20% off. An additional 5 dollars are taken off the sale price by using a discount coupon. A sales tax of 8% is paid on the final selling price. The total amount the shopper pays for the coat is
Suppose the estimated 20 billion dollar cost to send a person to the planet Mars is shared equally by the 250 million people in the U.S. Then each person's share is
Add the whole parts first, then bound the fractions.
Still stuck? Show hint 2 →
Hint 2 of 2
1β7 + 1β2 + 1β19 β at most 3β2, at least 1β2.
Show solution
Approach: split whole and fractional parts
Whole parts: 2 + 3 + 5 = 10. Fractions: 1β7 + 1β2 + 1β19 sits between 1β2 and 1 (clearly more than 1β2 since one term is already 1β2, but less than 1 since 1β7 + 1β19 < 1β2).
Total between 10 + 1β2 = 10 1β2 and 10 + 1 = 11. Answer B.
Joyce made 12 of her first 30 shots in the first three games of this basketball game, so her seasonal shooting average was 40%. In her next game, she took 10 shots and raised her seasonal shooting average to 50%. How many of these 10 shots did she make?
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Answer: E — 8.
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Hint 1 of 2
After the next game her total shots is 40 and average is 50%, so total made = 20.
Still stuck? Show hint 2 →
Hint 2 of 2
She'd already made 12 β how many more did she need?
Show solution
Approach: back out new made from new percent
After 40 shots at 50%, she's made 20. She had 12 before, so she made 20 β 12.
Half the people in a room left. One third of those remaining started to dance. There were then 12 people who were not dancing. The original number of people in the room was what?
Show answer
Answer: C — 36.
Show hints
Hint 1 of 2
After the leavers, half remain; of those, 2β3 are not dancing.
Still stuck? Show hint 2 →
Hint 2 of 2
(1β2) Γ (2β3) Γ N = 12.
Show solution
Approach: compose the two fractions
Non-dancers are 1β2 Γ 2β3 = 1β3 of the original group.
Sale prices at the Ajax Outlet Store are 50% below original prices. On Saturdays an additional discount of 20% off the sale price is given. What is the Saturday price of a coat whose original price is $180?
Show answer
Answer: B — $72.
Show hint
Hint 1
Apply each discount as a multiplier: 0.5 then 0.8.
Mika wants to estimate how far a new electric bike goes on a full charge. She made two trips totaling 40 miles: the first used 12 of the battery and the second used 310 of the battery. How many miles can the bike go on a fully charged battery?
Show answer
Answer: C — 50 miles.
Show hints
Hint 1 of 2
Add the two battery fractions to see what share of a full charge covered the 40 miles.
Still stuck? Show hint 2 →
Hint 2 of 2
Then scale up from that share to a whole battery.
Show solution
Approach: scale up from the fraction used
The two trips used Β½ + 3/10 = 4/5 of the battery for 40 miles.
A poll asked some people whether they liked solving mathematics problems, and exactly 74% answered "yes." What is the fewest possible number of people who could have been asked?
Show answer
Answer: D — 50 people.
Show hints
Hint 1 of 2
74% of the group must be a whole number of people.
Still stuck? Show hint 2 →
Hint 2 of 2
Reduce 74/100 to lowest terms; the denominator is the smallest possible group size.
Show solution
Approach: reduce the percentage to lowest terms
74% = 74/100 = 37/50 in lowest terms, so the number of people must be a multiple of 50.
Harold made a plum pie to take on a picnic. He was able to eat only 14 of the pie, and he left the rest for his friends. A moose came by and ate 13 of what Harold left behind. After that, a porcupine ate 13 of what the moose left behind. How much of the original pie still remained after the porcupine left?
Show answer
Answer: D — 1/3.
Show hints
Hint 1 of 2
Each eater leaves behind a fraction of what they found. Multiply those leftovers together.
A cup of boiling water (212°F) is placed to cool in a room whose temperature remains constant at 68°F. Suppose the difference between the water temperature and the room temperature is halved every 5 minutes. What is the water temperature, in degrees Fahrenheit, after 15 minutes?
Show answer
Answer: B — 86°F.
Show hints
Hint 1 of 2
Don't track the temperature directly — track the difference from room temp.
Still stuck? Show hint 2 →
Hint 2 of 2
Initial gap: 212 − 68 = 144. Halved three times in 15 minutes: 144 / 23 = 18. Add to room temp.
Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?
Show answer
Answer: E — 54%.
Show hints
Hint 1 of 2
Each transfer leaves a fraction behind. Multiply the "keep" fractions together.
Still stuck? Show hint 2 →
Hint 2 of 2
Keeps: 0.8 × 0.9 × 0.75.
Show solution
Approach: chain the leftover fractions
After Pedro: 80% remains. After Ebony: 90% of that. After Jimmy: 75% of what's left.
The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
Show answer
Answer: C — 12/7.
Show hints
Hint 1 of 2
Three steps: take reciprocals, average, take reciprocal again.
Still stuck? Show hint 2 →
Hint 2 of 2
Reciprocals: 1, 1/2, 1/4. Sum = 7/4. Average = 7/12. Final reciprocal: 12/7.
What is the correct ordering of the three numbers 519, 721, and 923, in increasing order?
Show answer
Answer: B — 5/19 < 7/21 < 9/23.
Show hint
Hint 1
Each fraction is of the form (n)/(n + 14) for n = 5, 7, 9. Write as 1 − 14/(n + 14). Bigger denominator on the subtracted piece ⇒ bigger overall fraction.
The taxi fare in Gotham City is $2.40 for the first 12 mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?
Show answer
Answer: C — 3.3 miles.
Show hint
Hint 1
Subtract the $2 tip from $10. Then subtract the $2.40 flag-drop for the first half-mile. Convert the rest at $0.20 per 0.1 mile = $2 per mile.
Show solution
Approach: peel off fixed costs, then divide the remainder
Available for fare: $10 − $2 = $8.
After the first 1/2 mile costing $2.40: $8 − $2.40 = $5.60 left.
Additional rate: $0.20 / 0.1 mile = $2 per mile. So $5.60 buys 2.80 miles.
There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5 : 4. There are 180 students at Winthrop Middle School, where the ratio of boys to girls is 4 : 5. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?
Show answer
Answer: C — 22/45.
Show hint
Hint 1
Each ratio uses 9 parts. Compute girls at each school, then total girls / total students.
Of the 500 balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed from the bag so that 75% of the remaining balls are red?
Show answer
Answer: D — 100 red balls.
Show hint
Hint 1
Blue balls don't change. 75% red ⇒ 25% blue, so the 100 blue balls represent 25% of the new total.
Show solution
Approach: blue stays constant, so use blue to find new total
Initial: 400 red, 100 blue.
After removal, 25% blue means total = 100 / 0.25 = 400 balls.
A jar contains five different colors of gumdrops: 30% are blue, 20% are brown, 15% red, 10% yellow, and the other 30 gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
Show answer
Answer: C — 42.
Show hints
Hint 1 of 2
Green = 100% − 30 − 20 − 15 − 10 = 25%, so 30 gumdrops = 25% ⇒ total = 120.
Still stuck? Show hint 2 →
Hint 2 of 2
Brown starts at 20% · 120 = 24. Add half the blue gumdrops (which switch color).
Show solution
Approach: find the total, then update brown
Green % = 25 ⇒ total = 30 / 0.25 = 120.
Blue count: 30% · 120 = 36. Brown count: 20% · 120 = 24.
A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less detergent than the large size. The large size contains twice as much detergent as the small size and costs 30% more than the medium size. Rank the three sizes from best to worst buy.
Show answer
Answer: E — MLS (best M, then L, then S).
Show hint
Hint 1
Set the small price to $1 and the large size to 10 oz. Derive the rest, then compute price per oz.
Show solution
Approach: anchor sizes, compute unit prices
Small: $1, 5 oz (large is twice the small ⇒ small = 5 oz).
Medium: $1.50 (50% more than small), 8 oz (20% less than 10 oz large).
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during that time she has used it for 60 minutes. If she doesn't talk any more but leaves the phone on, how many more hours will the battery last?
Show answer
Answer: B — 8 more hours.
Show hint
Hint 1
Idle uses 1/24 per hour; using uses 1/3 per hour. So far: 8 hr idle + 1 hr in use.
Show solution
Approach: battery fraction used so far
Used: 8 · (1/24) + 1 · (1/3) = 1/3 + 1/3 = 2/3.
Remaining: 1/3 of battery. At idle rate 1/24 per hour: time = (1/3) · 24 = 8 hr.
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?
Show answer
Answer: D — 27 people.
Show hint
Hint 1
1/4 of the chairs are empty = 6 ⇒ chairs total. Then 3/4 of chairs gives # seated people.
Show solution
Approach: find chairs, then people
Empty chairs (1/4 of total) = 6 ⇒ chairs = 24.
Seated people = (3/4)(24) = 18, which is 2/3 of all people.
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is 2/5. What fraction of the people in the room are married men?
Show answer
Answer: B — 3/8.
Show hint
Hint 1
Pick 5 women. 2 single, 3 married. Married women bring 3 husbands. Total people: 5 + 3 = 8.
Show solution
Approach: anchor with concrete count
Let women = 5. Single: 2; married: 3.
Married men = 3 (one per married woman). Total people: 5 + 3 = 8.
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by 10%. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40 on Thursday?
Show answer
Answer: B — $39.60.
Show hints
Hint 1 of 2
A 10% increase followed by a 10% decrease is not a wash — the cut is taken off a bigger number.
Still stuck? Show hint 2 →
Hint 2 of 2
Multiply the two factors: ×1.1 then ×0.9.
Show solution
Approach: chain the two percent changes
Friday: 40 × 1.1 = 44. Monday: 44 × 0.9 = 39.60.
Equivalently 40 × 1.1 × 0.9 = 40 × 0.99 = 39.60 — a hair under the original.
The students in Mrs. Sawyer's class each chose one of five kinds of candy in a taste test. The bar graph shows their preferences. What percent of her class chose candy E?
Show answer
Answer: E — 20%.
Show hints
Hint 1 of 2
First find how many students there are in all.
Still stuck? Show hint 2 →
Hint 2 of 2
With 25 students total, each one is 4% of the class.
Show solution
Approach: part over whole, then turn into a percent
The class total is 6 + 8 + 4 + 2 + 5 = 25 students.
Candy E was chosen by 5 of them: 5/25 = 1/5 = 20%.
Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?
Show answer
Answer: E — 55 inches.
Show hints
Hint 1 of 2
Find the common starting height from Shea's 20% growth.
Still stuck? Show hint 2 →
Hint 2 of 2
Ara grew half as many inches as Shea β match inches, not percents.
Show solution
Approach: back out the starting height, then add Ara's inches
Shea grew 20% to reach 60, so the start was 60 Γ· 1.2 = 50 inches, meaning Shea grew 10 inches.
Ara grew half that, 5 inches, ending at 50 + 5 = 55 inches.
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 answer
Answer: E — Milepost 130.
Show hints
Hint 1 of 2
The two exits are 160 β 40 = 120 mileposts apart.
Still stuck? Show hint 2 →
Hint 2 of 2
Go three-fourths of that distance past milepost 40.
Show solution
Approach: fraction of the gap, added to the start
From milepost 40 to 160 is 120 miles, and three-fourths of 120 is 90.
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?
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Answer: B — 27%.
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Hint 1 of 2
Turn the ratio into parts: 11 won and 4 lost make 15 games in all.
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Hint 2 of 2
The lost fraction is 4 out of 15.
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Approach: ratio β total parts β percent
Treat the ratio as 11 wins and 4 losses, so 15 games total.
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?
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Answer: B — 5 more.
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Hint 1 of 2
Count how many she actually got right in each subject.
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Hint 2 of 2
Compare that total to 60% of all 75 problems.
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Approach: actual correct vs. the 60% target
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 Γ 75 = 45 correct, so she was short by 5.
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?
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Answer: E — 11 recipes.
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Hint 1 of 2
A 25% drop leaves three-fourths of the 108 students.
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Hint 2 of 2
Find their cookies, then round up to whole pans of 15.
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Approach: fewer guests β cookies β round up pans
Three-fourths of 108 is 81 students, eating 81 Γ 2 = 162 cookies.
For a sale, a store owner reduces the price of a $10 scarf by 20%. Later the price is lowered again, this time by one-half of the reduced price. The price is now
At Annville Junior High School, 30% of the students in the Math Club are in the Science Club, and 80% of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club?
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Answer: E — 40 students.
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Hint 1 of 2
Find how many students are in both clubs, using the Science Club side.
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Hint 2 of 2
That same number is 30% of the Math Club.
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Approach: find the overlap, then back out the whole
Both clubs share 80% of 15 = 12 students.
Those 12 make up 30% of the Math Club, so the Math Club has 12 Γ· 0.3 = 40 students.
When Walter drove up to the gasoline pump, his tank was 1/8 full. He bought 7.5 gallons, after which the tank was 5/8 full. How many gallons does the tank hold when it is full?
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Answer: D — 15 gallons.
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Hint 1 of 2
The 7.5 gallons raised the level from 1/8 to 5/8 β what fraction of the tank is that?
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Hint 2 of 2
Then scale up to a full tank.
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Approach: fraction added gives the whole
Going from 1/8 to 5/8 is 4/8 = half the tank, filled by 7.5 gallons.
In the fall of 1996, 800 students took part in an annual school clean-up day. The organizers expect that in each of 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants expected in the fall of 1999 is
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Answer: E — 2700.
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Hint 1 of 2
A 50% increase multiplies by 1.5 each year.
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Hint 2 of 2
Apply that three times to reach 1999.
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Approach: multiply by 1.5 three times
Three yearly increases multiply by 1.5 three times: 800 Γ (3/2)Β³ = 800 Γ 27/8.
Ana's monthly salary was $2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was
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Answer: A — 1920 dollars.
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Hint 1 of 2
A 20% raise then a 20% cut do NOT cancel out.
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Hint 2 of 2
Multiply by 1.2, then by 0.8.
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Approach: apply both percent changes in turn
After the raise: 2000 Γ 1.2 = 2400. After the cut: 2400 Γ 0.8 = 1920.
At Clover View Junior High, half of the students go home on the school bus, one fourth go home by automobile, and one tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home?
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Answer: B — 3/20.
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Hint 1 of 2
Add the three known fractions over a common denominator (20).
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Hint 2 of 2
Subtract that from 1.
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Approach: subtract the known fractions from the whole
A team won 40 of its first 50 games. How many of the remaining 40 games must this team win so that it will have won exactly 70% of its games for the season?
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Answer: B — 23.
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Hint 1 of 2
The season has 50 + 40 = 90 games; find 70% of that.
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Hint 2 of 2
Subtract the 40 wins already in hand.
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Approach: target total wins, then subtract
70% of the 90 games is 63 wins needed for the season.
Already having 40, the team must win 63 β 40 = 23 more.
During the softball season, Judy had 35 hits. Among her hits were 1 home run, 1 triple, and 5 doubles. The rest of her hits were singles. What percent of her hits were singles?
A dress originally priced at 80 dollars was put on sale for 25% off. If 10% tax was added to the sale price, then the total selling price (in dollars) of the dress was
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Answer: D — 66 dollars.
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Hint 1 of 2
Take 25% off first, then add the tax on the reduced price.
A bag contains only blue balls and green balls. There are 6 blue balls. If the probability of drawing a blue ball at random from this bag is 14, then the number of green balls in the bag is
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Answer: B — 18.
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Hint 1 of 2
If blue is 1/4 of the balls, the total is 4 times the blue count.
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Hint 2 of 2
Subtract the blue balls to get the green ones.
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Approach: find the total, then subtract blue
Blue = 1/4 of all, so the total is 4 Γ 6 = 24 balls.
Betty used a calculator to find the product 0.075 Γ 2.56. She forgot to enter the decimal points. The calculator showed 19200. If Betty had entered the decimal points correctly, the answer would have been
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Answer: B — .192.
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Hint 1 of 2
Count the total number of digits after the decimal point in the two factors.
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Hint 2 of 2
Move the decimal point in 19200 that many places to the left.
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Approach: count decimal places
0.075 has 3 decimal places, 2.56 has 2 β total 5. Move the decimal in 19200 five places to the left.
Steph scored 15 baskets out of 20 attempts in the first half of a game, and 10 baskets out of 10 attempts in the second half. Candace took 12 attempts in the first half and 18 attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
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Answer: C — 9 more baskets.
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Hint 1 of 2
Steph took 20 + 10 = 30 attempts; Candace took 12 + 18 = 30. Same total, same overall % ⇒ same total makes.
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Hint 2 of 2
Steph: 15 + 10 = 25 makes. Candace must also make 25. With her per-half percentages strictly below Steph's, only one (f, s) split works.
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Approach: match totals, then narrow by per-half constraints
Same total attempts (30 each) + same overall percentage ⇒ same total makes. Steph made 15 + 10 = 25, so Candace also made 25.
Let Candace's makes be f (first half, out of 12) and s (second, out of 18). Per-half percentages strictly below Steph: f/12 < 15/20 = 3/4 ⇒ f ≤ 8. And s/18 < 1 ⇒ s ≤ 17.
f + s = 25, f ≤ 8, s ≤ 17 ⇒ only f = 8, s = 17 fits.
A store increased the original price of a shirt by a certain percent and then decreased the new price by the same amount. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased?
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Answer: E — 40%.
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Hint 1 of 2
Multiplying by (1+p) then (1−p) gives 1 − p2. That equals 0.84.
One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, and so on. After how many pourings does exactly one tenth of the original water remain?
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Answer: D — 9.
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Hint 1 of 2
After each pouring, multiply by what's left: 1/2, then 2/3, then 3/4, β¦
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Hint 2 of 2
These products telescope to a simple fraction.
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Approach: multiply the surviving fractions (they telescope)
After k pourings the remaining fraction is Β½ Β· β Β· ΒΎ Β· β¦ Β· k/(k+1), which telescopes to 1/(k+1).
Jack had a bag of 128 apples. He sold 25% of them to Jill. Next he sold 25% of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?
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Answer: D — 71.
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Hint 1 of 2
Selling 25% means keeping 75% = 3β4.
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Hint 2 of 2
Multiply by 3β4 twice, then subtract 1 for the apple given to the teacher.
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Approach: keep ΒΎ twice, then subtract 1
After Jill: 128 Γ 3β4 = 96. After June: 96 Γ 3β4 = 72.
Tom's Hat Shoppe increased all original prices by 25%. Now the shoppe is having a sale where all prices are 20% off these increased prices. Which statement best describes the sale price of an item?
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Answer: E — The sale price is the same as the original price.
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Hint 1 of 2
Apply each percent change as a multiplier and multiply them.
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Hint 2 of 2
1.25 Γ 0.80 β what does that equal?
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Approach: multiply the two factors
+25% means Γ1.25; β20% off means Γ0.80. Combined: 1.25 Γ 0.80 = 1.00.
