Question

luke is thinking of a number twice his age. the sum of the digits is 8. if the ones digit is 3 times larger than the tens digit, how old is luke?

Explanation:

Step1: Define variables for the number

Let the tens digit of the number be \( x \) and the ones digit be \( y \). The number can be expressed as \( 10x + y \).

Step2: Use the digit sum condition

We know that the sum of the digits is 8, so \( x + y = 8 \).

Step3: Use the ones digit - tens digit condition

The ones digit is 3 times larger than the tens digit, which means \( y = 3x \) (note: "3 times larger" here is interpreted as \( y = 3x \); if it were "3 times as large", it would also be \( y = 3x \) in this context).

Step4: Solve the system of equations

Substitute \( y = 3x \) into \( x + y = 8 \):
\( x+3x = 8 \)
\( 4x = 8 \)
\( x = 2 \)
Then \( y = 3x = 3\times2 = 6 \), so the number is \( 10x + y = 10\times2 + 6 = 26 \).

Step5: Find Luke's age

Since the number is twice Luke's age, Luke's age is \( \frac{26}{2}=13 \).

Answer:

Luke is 13 years old.