Lower Elementary:
Question: Korey earned twice as much as Ronnie. Ronnie earned twice as much as Jacob. If Jacob earned $24, how much did Korey earn?
Answer: $96
Solution: Ronnie earned twice as much as Jacob, so Ronnie earned $48 (24 doubled = 20 doubled + 4 doubled = 40 + 8 = 48). Korey earned twice as much as Ronnie, so Korey earned $96 (48 doubled = 40 doubled + 8 doubled = 80 + 16 = 96).
Upper Elementary:
Question: A middle school hockey game is 3 periods and each period is 15 minutes long. If the game is half way through the second period, how much of the game is left?
Answer: 22 minutes and 30 seconds
Solution: Let’s find how long the entire game is. Since there are 3 periods and each period is 15 minutes, the entire game is 45 minutes (15 × 3 = 45). After the first period, 15 minutes have passed, so now there is 30 minutes left in the game. Half of 15 is 7 1/2 (half of 15 = half of 10 + half of 5 = 5 + 2 1/2 = 7 1/2). So, there are 22 and a half minutes, or 22 minutes and 30 seconds left in the game (30 – 7 1/2 = 22 1/2).
Middle School:
Question: Three friends worked together to pay for new speakers. Tristan paid $100, Eddie paid $25, and Nigel paid $75. They then resold them for $240. How should they split the money earned so that it is fair?
Answer: Tristan: $120, Eddie: $30, Nigel: $90
Solution: First, we need to find the total amount the speakers cost. 100 + 25 + 75 = 200. Now, let’s find the fractional part that each person paid. Tristan paid 1/2 of the amount (100/200 = 1/2). Eddie paid 1/8 of the amount (25/200 = 1/8). Nigel paid 3/8 of the amount (75/200 = 3/8). Since they resold it for $240, we can find the amount that each person should get. 1/2 of 240 is 120, so Tristan gets $120. 1/8 of 240 is 30, so Eddie gets $30. 3/8 of 240 is 90, so Nigel gets $90.
Algebra and Up:
Question: Where do the graphs y = x – 3 and x2 + y2 = 9 intersect?
Answer: (0, –3) and (3, 0)
Solution: One way to solve this problem is to substitute one function into the other. If we substitute y = x – 3 into x2 + y2 = 9, we have:
x2 + (x – 3)2 = 9
Expand (x – 3)2.
x2 + x2 – 6x + 9 = 9
Combine like terms.
2x2 – 6x + 9 = 9
Subtract 9 from both sides.
2x2 – 6x = 0
Factor a 2x from each term.
2x(x – 3) = 0
The two values of x are x = 0 and x = 3. To find the y values, we substitute the x values into either of the original equations.
x = 0
y = 0 – 3 = –3
This point is (0, –3)
x = 3
y = 3 – 3 = 0
This point is (3, 0)
The two points where the graphs intersect are at (0 –3) and (3, 0).
