# Solutions

Excellent! Are you ready to check your child’s answer? Look below to see if their solution matches ours.

**Lower Elementary:**

*Answer:*22 minutes older

*Solution:*The date changes from December 17 to December 18 at 12:00 AM. There are 60 minutes in an hour, and 45 minutes is 15 minutes away from 60 minutes, so Margie was born 15 minutes before midnight. 7 minutes later is when Carrie was born, so Margie is 15 minutes + 7 minutes = 22 minutes older than Carrie.

**Upper Elementary:**

*Answer:*23 years old

*Solution:*In order to compare the fractional parts, we need to give them the same denominator – in this case, 2 × 3 = 6. 1/2 × 3/3 = 3/6 of the candles are blue, and 1/3 × 2/2 = 2/6 of the candles are green. This means that 3/6 + 2/6 = 5/6 of the candles are blue or green, leaving 1 – 5/6 = 1/6 of the candles yellow. If 1/6 of the candles is the same as 4 candles, then the whole number of candles must be 6 × 4 = 24. If there is 1 more candle on the cake than the number of years Barb has been alive, then she is 24 – 1 = 23 years old.

**Middle School:**

*Answer:*0.29%

*Solution:*The probability that the first lollipop Jenny picks will be blue raspberry-flavored is 8 lollipops out of 50 total lollipops = 8/50 = 4/25. If she picks this flavor, then the probability of the second lollipop being the same flavor is 8 – 1/50 – 1 = 7/49 = 1/7. If she picks this flavor for the first and the second, then the probability of the third lollipop being the same flavor is 7 – 1/49 – 1 = 6/48 = 1/8. The probability of all of these events occurring is 4/25 × 1/7 × 1/8 = 4/1,400 ≈ 0.29%.

**Algebra and Up:**

*Answer:*41 minutes

*Solution:*We need to combine Nick’s, Rose’s, and Leif’s decorating rates by addition. Choosing a common denominator, Nick’s rate is 1/1.5 = 28/42, Rose’s is 1/2 = 21/42, and Leif’s is 1/3.5 = 12/42. Their combined rate of decorating is 28/42 + 21/42 + 12/42 = 61/42, or 61 parties in 42 minutes. To find their combined rate for 1 party, we set 61/42 equal to 1/x (1 party in x minutes). Using cross products, we find that x = 42/61 of an hour = 42/61 of 60 minutes = 42/61 × 60 minutes ≈ 41 minutes.