January 22, 2018

Cara was born on January 1, 2010, and her mother, Sydney, was born on January 1, 1982. In what year will Sydney’s age be twice Cara’s age?

If we let Cara’s age be C, then Sydney’s age is S = C + 28, since Cara was born 28 years after Sydney. We are interested in determining when Sydney’s age is twice Cara’s age, in other words, when S = 2C. Substituting, we have C + 28 = 2C. Solving for C we get C = 28. Therefore, Sydney’s age will be twice Cara’s age in the year 2010 + 28 = 2038.

 

Cara’s brother, Nile, celebrated a birthday this year (2018) on January 4th. Cara’s current age is 4/5 Nile’s current age. How old was Nile’s mom, Sydney, when he was born?

Cara’s current age is 2018 – 2010 = 8 years old. We are told that Cara’s current age is 4/5 Nile’s current age, N. That means (4/5)N = 8. Solving for N, we see that Nile’s current age is N = (5/4)8 = 10 years old. Sydney’s current age, S, is 2018 – 1982 = 36 years. Therefore, ten years ago, when Nile was born, Sydney’s age was 36 – 10 = 26 years old.

 

The sum of the ages of Cara, Nile and Sydney each year forms an arithmetic progression. The sum of their ages in 2018 is 54. In what year will the sum of their ages be 78?

From the previous problem we know that Cara’s, Nile’s and Sydney’s current ages are 8, 10 and 36, respectively. The sum of these ages is 8 + 10 + 36 = 54. Next year the sum of their ages will be 9 + 11 + 37 = 57. The following year the sum of their ages will be10 + 12 + 38 = 60. Notice that the common difference in the arithmetic progression is +3 since each year the ages of Cara, Nile and Sydney each increase by 1. To determine in how many years the sum of the ages will be 78, we can find out how many times 3 is added to 54 to get to 78. In other words, we can solve the equation 54 + 3x = 78 ® 3x = 24 ® x = 8. So, the sum of the ages of Cara, Nile and Sydney will be 78 in the year 2018 + 8 = 2026.

 

Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2
contains ONLY PROBLEMS.