1/3 of a number is 6 more than 1/6 of that number. What is the number?

Concept:

To find an unknown number from a relationship between two fractional parts of it, translate the statement directly into a linear equation in one variable, then combine the fractional coefficients of the variable and solve for it.

Working:

1. Let the number be P. The statement translates to: (1/3)P = 6 + (1/6)P.

2. Subtract (1/6)P from both sides to collect the P terms on one side: (1/3)P − (1/6)P = 6.

3. Combine the fractional coefficients on the left: (1/3) − (1/6) = (2/6) − (1/6) = (1/6), so (1/6)P = 6.

4. Multiply both sides by 6 to isolate P: P = 36.

Check:

With P = 36: 1/3 of 36 is 12, and 1/6 of 36 is 6. Since 12 is exactly 6 more than 6, the relationship in the statement holds, confirming P = 36.