Track / Stage 5 / 305 of 400 #905 of 2000

Problem 905

AIME late

Algebra Difficulty 5.7 Find the answer

9. (10 points) The Fifth Plenary Session of the 18th Central Committee of the Communist Party of China, which concluded on October 29, 2015, determined the policy allowing universal second children. Xiao Xiao's father, after seeing the news on that day, told Xiao Xiao: The total age of our family this year is 7 times your age. If we add a younger brother or sister next year, the total age of our family in 2020 will be 6 times your age at that time. How old is Xiao Xiao this year? $\qquad$ years old.

Your answer

Official solution

【Analysis】From 2015 to 2020, 5 years have passed. Since Xiao Xiao's brother will be born next year (2016), the total age of the family (4 people) increased by $5 \times 3 + 4 = 19$ (years), while Xiao Xiao grew 5 years older. If in 2020, the total age of the family is still 7 times that of Xiao Xiao, then the family's age should have increased by $5 \times 7 = 35$ years, but in fact, it only increased by 19 years, which is $35 - 19 = 16$ years less, which is $7 - 6 = 1$ times less. Therefore, in 2020, Xiao Xiao's age is $16 \div 1 = 16$ years old, so Xiao Xiao's age this year is $16 - 5 = 11$ years old.

【Solution】Solution: $2020 - 2015 = 5$ (years)
$\begin{array}{l} (5 \times 7 - 5 \times 3 - 4) \div (7 - 6) \\ =(35 - 15 - 4) \div 1 \\ =(20 - 4) \div 1 \\ = 16 \div 1 \\ = 16 \text { (years) } \\ 16 - 5 = 11 \text { (years) } \end{array}$

Therefore, the answer is: 11 years old.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.