Julie is preparing a speech. It must last between one-half hour and three-quarters of an hour, and her ideal rate is 150 words per minute. If she speaks at that rate, which of the following word counts is an appropriate length?
Show hint (soft nudge)
Turn each time limit into a word count at 150 words per minute.
Show hint (sharpest)
The answer must fall between those two counts.
Show solution
At 150 words per minute, 30 minutes is 4500 words and 45 minutes is 6750 words.
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.
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?
Show hint (soft nudge)
Count the yellow beans in each bag, then total them.
Show hint (sharpest)
Divide the yellow total by the grand total of beans.
Show solution
Yellow beans: 13 + 7 + 6 = 26, out of 26 + 28 + 30 = 84 beans.
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.
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 hint (soft nudge)
Track each $100 through both years.
Show hint (sharpest)
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 hint (soft nudge)
Find the price of one box before and during the sale.
Show hint (sharpest)
Percent decrease compares the drop to the original price.
Show solution
A box cost $5 ÷ 4 = $1.25 before and $4 ÷ 5 = $0.80 on sale.
The decrease is $0.45 ÷ $1.25 = 36%, closest to 35%.
A pair of 8-sided dice have sides numbered 1 through 8, each equally likely. What is the probability that the product of the two numbers facing up exceeds 36?
Show hint (soft nudge)
A big product needs big rolls — case on each high value of one die.
Show hint (sharpest)
Count the ordered pairs whose product is more than 36 out of all 64.
Show solution
Going through the high rolls: a 5 needs an 8 (1 way), a 6 needs 7–8 (2), a 7 needs 6–8 (3), and an 8 needs 5–8 (4).
That's 1 + 2 + 3 + 4 = 10 ordered pairs out of 64, a probability of 10/64 = 5/32.
Some positive integers have both properties: (I) the sum of the squares of their digits is 50, and (II) each digit is larger than the one to its left. The product of the digits of the largest such integer is
Show hint (soft nudge)
The digits strictly increase, so they're distinct; squaring shows there can be at most four of them.
Show hint (sharpest)
To make the number large, push the leading digits up while keeping the square-sum at 50.
Show solution
Five increasing digits would have squares summing to at least 1+4+9+16+25 = 55 > 50, so at most four digits.
The digits 1, 2, 3, 6 give 1 + 4 + 9 + 36 = 50 and form the largest such number, 1236.
