Question
The sum of the ages of two cousins is 46. The younger cousin is 10 more than a third of the older cousin’s age. How old is the younger cousin?
Responses
A 1919
B 2727
C 1818
D 46
Answers
GPT-4o mini
Let's define the ages of the cousins:
Let \( x \) be the age of the older cousin and \( y \) be the age of the younger cousin.
From the information given, we have the following two equations:
1. \( x + y = 46 \) (the sum of their ages)
2. \( y = \frac{1}{3}x + 10 \) (the younger cousin's age in relation to the older cousin's age)
Now, we can solve these equations step by step.
Substitute Equation 2 into Equation 1:
\[
x + \left(\frac{1}{3}x + 10\right) = 46
\]
Combine like terms:
\[
x + \frac{1}{3}x + 10 = 46
\]
To combine \( x \) and \( \frac{1}{3}x \), rewrite \( x \) as \( \frac{3}{3}x \):
\[
\frac{3}{3}x + \frac{1}{3}x + 10 = 46
\]
This gives:
\[
\frac{4}{3}x + 10 = 46
\]
Now, subtract 10 from both sides:
\[
\frac{4}{3}x = 36
\]
Next, multiply both sides by \( \frac{3}{4} \) to solve for \( x \):
\[
x = 36 \cdot \frac{3}{4} = 27
\]
Now that we have \( x \), we can find \( y \) using Equation 1:
\[
y = 46 - x = 46 - 27 = 19
\]
Thus, the younger cousin is **19 years old**.
Now let's refer back to your responses:
A: 1919
B: 2727
C: 1818
D: 46
None of the responses match 19. It seems like there might be a misunderstanding in the answer options provided. However, the solution to the problem indicates that the correct age of the younger cousin is **19**.
Let \( x \) be the age of the older cousin and \( y \) be the age of the younger cousin.
From the information given, we have the following two equations:
1. \( x + y = 46 \) (the sum of their ages)
2. \( y = \frac{1}{3}x + 10 \) (the younger cousin's age in relation to the older cousin's age)
Now, we can solve these equations step by step.
Substitute Equation 2 into Equation 1:
\[
x + \left(\frac{1}{3}x + 10\right) = 46
\]
Combine like terms:
\[
x + \frac{1}{3}x + 10 = 46
\]
To combine \( x \) and \( \frac{1}{3}x \), rewrite \( x \) as \( \frac{3}{3}x \):
\[
\frac{3}{3}x + \frac{1}{3}x + 10 = 46
\]
This gives:
\[
\frac{4}{3}x + 10 = 46
\]
Now, subtract 10 from both sides:
\[
\frac{4}{3}x = 36
\]
Next, multiply both sides by \( \frac{3}{4} \) to solve for \( x \):
\[
x = 36 \cdot \frac{3}{4} = 27
\]
Now that we have \( x \), we can find \( y \) using Equation 1:
\[
y = 46 - x = 46 - 27 = 19
\]
Thus, the younger cousin is **19 years old**.
Now let's refer back to your responses:
A: 1919
B: 2727
C: 1818
D: 46
None of the responses match 19. It seems like there might be a misunderstanding in the answer options provided. However, the solution to the problem indicates that the correct age of the younger cousin is **19**.