Lower Elementary:
Question: Kelsey bought a pack of 92 balloons. She used half of them for decorations for her friend’s birthday party. She then used half of the remaining balloons for a game. How many balloons does Kelsey have left?
Answer: 23 balloons
Solution: Half of 92 is the same as half of 90 + half of 2. Half of 90 is 45 and half of 2 is 1. 45 + 1 = 46. Half of 92 is 46. Kelsey used 46 balloons to decorate. Then she uses half of the remaining for a game. Half of 46 is 23. She used 23 for the balloon game, which means that she has 23 balloons left.
Upper Elementary:
Question: Korey is selling tickets for a concert. He sold 500 tickets and for each ticket he makes a $10 profit. Korey gets to keep 1/4 of the profits. How much of the profit does Korey get to keep?
Answer: $1,250
Solution: First, find the total amount of money earned from selling the tickets. To do this, multiply the number of tickets sold by the cost of each ticket. 500 × 10 = 5,000. The total money earned is $5,000. To see how much of the profit Korey keeps, take 1/4 of $5,000. 1/4 of a number is the same as taking half of a half. Half of 5,000 is 2,500 and half of 2,500 is 1,250. Korey gets to keep $1,250 of the profit.
Middle School:
Question: Walter’s gas tank reads that it is 25% full. He puts in 15 gallons and now completely full. What is the maximum amount of gas that Walter’s tank can hold?
Answer: 20 gallons
Solution: Because the tank went from 25% full to completely, or 100% full when 15 gallons were added, this means that 15 gallons is 75% of the total tank (100% – 25% = 75%). So now we ask the question “75% of what number is 15?” 75% means the same as 3/4. Another way of reading this problem is “3/4 of what number is 15?” To do this, break 15 into 3 equal parts, and then multiply by 4. We break it into 3 parts because that will get us the value of 1/4 of the tank, and then we multiply that by 4 to get the whole tank. 15 ÷ 3 = 5. 5 × 4 = 20. Walter’s gas tank can hold a maximum of 20 gallons.
Algebra and Up:
Question: Which two integers have a product of 315 and a sum of 36?
Answer: 15 and 21
Solution: Let x represent one of the integers and let y represent the other integer. Rewriting the question in terms of the x and y gives us the equations:
xy = 315
x + y = 36
One way to solve this is to use substitution. Using the bottom equation to solve for y, we have:
y = 36 – x
Substitute that value into the top equation to get:
x(36 – x) = 315
Now we solve for x:
36x – x2 = 315
0 = x2 – 36x + 315
0 = (x – 15)(x – 21)
x = 15 or x = 21
Now we plug in each value into y = 36 – x to figure out the value of y.
y = 36 – 15 = 21
y = 36 – 21 = 15
Notice that in either case, the two solutions are the same. The two integers that have a product of 315 and a sum of 36 are 15 and 21.
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