# Problem of the Week

Jul 4, 2021 | Folsom

# Did you figure it out?

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.

# Did you figure it out?

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.