Lower Elementary:
Question: Trisha had a bag of grapes that she ate from throughout the day. She ate half of the grapes at lunch. Then she ate half of what was left during her walk home. When she arrived home she had 8 grapes left. How many grapes did Trisha start with?
Answer: 32 grapes
Solution: To solve this problem we will work backwards. When Trisha arrived home she had 8 grapes. That means that she had double that amount before she started walking home. 8 doubled = 16. This means she had 16 after lunch. To find out how many she had before lunch, we double 16. To double 16, we can break it down into 10 and 6, double each part, and then add the parts together. 16 doubled = 10 doubled + 6 doubled = 20 + 12 = 32. Trisha started with 32 grapes.
Upper Elementary:
Question: Half of Jon’s music collection is hip-hop songs. A third of it is R&B songs. The remaining 24 songs are pop songs. How many songs of Jon’s music are hip-hop?
Answer: 72 songs
Solution: One way to solve this problem is to start by finding the fractional part of his music collection that is pop songs. To find it, take the whole (1) and subtract the sum of the other fractional parts. 1 – 1/2 – 1/3 = 1 – 3/6 – 2/6 = 1/6. 1/6 of Jon’s music is pop songs. That means that 1/6 represents 24 songs. Since 6 sixths make a whole, to find the total number of songs in his collection, multiply the number of pop songs by 6. 24 × 6 = 144. Jon has 144 total songs in his music collection. Half of them are hip-hop songs, so there are 72 hip-hop songs in his collection (half of 144 = half of 100 + half of 44 = 50 + 22 = 72).
Middle School:
Question: Bridgette makes 2% commission on each motorcycle that she sells. In June, she sold 16 motorcycles for $5,000 each. How much in commission does Bridgette make in commission in the month of June?
Answer: $1,600
Solution: One way to solve this problem is to find the commission that Bridgette makes on 1 motorcycle and then multiply it by 16, the number of motorcycles that she sold. Percent means “for each hundred”. There are fifty 100’s in 5,000, so 2% of 5,000 is 50 × 2 = 100. Bridgette earns $100 on commission for each motorcycle. Multiply by 16 to get the total commission she will earn for June. 16 × 100 = 1,600. She earned $1,600 in commission.
Algebra and Up:
Question: Taylor has a 12 inch stick. He is going to break it into 3 pieces. Each piece will have a length in inches that is a whole number. He wants at least 2 of the pieces to be of equal length. He will then construct a triangle out of the pieces. What are the possible lengths of the 3 pieces?
Answer: 4, 4, 4 and 5, 5, 2
Solution: He is breaking the stick into 3 pieces, at least 2 of the pieces have to be the same length, and the length of each piece has to be a whole number. Listing all the possible arrangements, we have:
1, 1, 10
2, 2, 8
3, 3, 6
4, 4, 4
5, 5, 2
To find out which ones make a triangle, we use the fact that for any triangle the sum of 2 sides must be greater than the sum of the 3rd side. One way to quickly check this is to add the 2 smaller sides together and see if it is larger than the longest side.
1 + 1 = 2 < 10 2 + 2 = 4 < 8 3 + 3 = 6 = 6 4 + 4 = 8 > 4
5 + 5 = 10 > 2
There only two combinations where the sum of the smaller pieces is greater than the longest piece are 4, 4, 4 and 5, 5, 2.
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