The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome?
Show hint (soft nudge)
Keep the thousands digit at 2, so the next palindrome looks like 2 _ _ 2.
Show hint (sharpest)
For that to mirror, the two middle digits must be equal — make them as small as you can while passing 2002.
Show solution
Don't change the leading 2, so the year stays of the form 2 _ _ 2 with equal middle digits.
The smallest such year after 2002 is 2112, whose digit product is 2 × 1 × 1 × 2 = 4.
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?
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First find how many students there are in all.
Show hint (sharpest)
With 25 students total, each one is 4% of the class.
Show solution
The class total is 6 + 8 + 4 + 2 + 5 = 25 students.
Candy E was chosen by 5 of them: 5/25 = 1/5 = 20%.
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¢.
A sequence of squares is made of identical square tiles. Each square's edge is one tile longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?
Show hint (soft nudge)
A square that is n tiles on a side uses n × n tiles.
Show hint (sharpest)
Compare 7 × 7 with 6 × 6.
Show solution
A square that is n tiles on a side uses n² tiles.
So the seventh needs 7² = 49 and the sixth needs 6² = 36: the difference is 49 − 36 = 13.
C
13.
Another way: difference of squares (no squaring needed)
A board game spinner is divided into three regions labeled A, B, and C. The probability the arrow stops on region A is 13 and on region B is 12. What is the probability the arrow stops on region C?
Show hint (soft nudge)
The three probabilities have to add up to 1.
Show hint (sharpest)
So region C gets whatever is left after A and B.
Show solution
Since the arrow must land somewhere, P(C) = 1 − 13 − 12.
For his birthday, Bert gets a box that holds 125 jellybeans when filled to capacity. A few weeks later, Carrie gets a larger box full of jellybeans. Her box is twice as high, twice as wide, and twice as long as Bert's. Approximately how many jellybeans did Carrie get?
Show hint (soft nudge)
Doubling every dimension does more than double the box — picture stacking copies of the small box inside the big one.
Show hint (sharpest)
Twice as long, wide, and high multiplies the volume by 2 × 2 × 2.
Show solution
Doubling all three dimensions multiplies the volume by 2 × 2 × 2 = 8.
So Carrie's box holds about 8 × 125 = 1000 jellybeans.
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 hint (soft nudge)
Discounts don't simply add — track the fraction of the price you still pay.
Show hint (sharpest)
After 30% off you pay 0.7 of the price; the next 20% off pays 0.8 of that.
Show solution
After 30% off you pay 0.70 of the original; taking another 20% off pays 0.80 of that.
So you pay 0.70 × 0.80 = 0.56 of the original — a 44% total discount, not 50%.
In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have?
Show hint (soft nudge)
Imagine she got all 10 right first, then see what each wrong answer costs.
Show hint (sharpest)
Turning one correct answer into a wrong one drops the score by 5 + 2 = 7.
Show solution
All 10 correct would score 5 × 10 = 50.
Each wrong answer (instead of right) costs 5 + 2 = 7 points, and 50 − 29 = 21 = 3 × 7.
So 3 were wrong and 7 were correct.
C
7.
Another way: set up an equation
Let x = number correct, so 10 − x are wrong: 5x − 2(10 − x) = 29.
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?
Show hint (soft nudge)
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.
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?
Show hint (soft nudge)
Work out how much juice one pear gives and how much one orange gives.
Show hint (sharpest)
With equal numbers of each fruit, just compare those per-fruit yields.
Show solution
One pear gives 8/3 oz of juice; one orange gives 8/2 = 4 oz.
Equal numbers of each fruit make the pear-to-orange juice ratio 8/3 : 4 = 2 : 3.
Pear's share of the blend is 2 ÷ (2 + 3) = 2/5 = 40%.
Loki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money, and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?
Show hint (soft nudge)
The three gifts are equal — pick a convenient size for that common gift, like $1.
Show hint (sharpest)
Then Moe started with $5, Loki $4, Nick $3; passing money around never changes the group total.
Show solution
Let each gift be $1. Since $1 is Moe's fifth, Loki's fourth, and Nick's third, they began with $5, $4, and $3.
Handing money over doesn't change the total: $5 + $4 + $3 = $12, and Ott now holds the three gifts, $3.
