Question

What number is such that one quarter of the difference between it and 4 is the same as one sixth of the difference between it and 3

Answer 1

Let the unknown number be \( x \).

We need to set up the equation based on the problem statement. According to the problem:

One quarter of the difference between \( x \) and 4 is expressed as:
\[
\frac{1}{4}(x - 4)
\]

One sixth of the difference between \( x \) and 3 is expressed as:
\[
\frac{1}{6}(x - 3)
\]

We want to find \( x \) such that these two expressions are equal:
\[
\frac{1}{4}(x - 4) = \frac{1}{6}(x - 3)
\]

To eliminate the fractions, we can find a common denominator, which is 12. Multiplying both sides of the equation by 12 gives:
\[
12 \cdot \frac{1}{4}(x - 4) = 12 \cdot \frac{1}{6}(x - 3)
\]

This simplifies to:
\[
3(x - 4) = 2(x - 3)
\]

Now, distribute on both sides:
\[
3x - 12 = 2x - 6
\]

Next, we will isolate \( x \) by subtracting \( 2x \) from both sides:
\[
3x - 2x - 12 = -6
\]
\[
x - 12 = -6
\]

Now, add 12 to both sides:
\[
x = 6
\]

Thus, the number is \( \boxed{6} \).

To verify, we can substitute \( x = 6 \) back into the original expressions:
1. One quarter of the difference between 6 and 4:
\[
\frac{1}{4}(6 - 4) = \frac{1}{4}(2) = \frac{1}{2}
\]

2. One sixth of the difference between 6 and 3:
\[
\frac{1}{6}(6 - 3) = \frac{1}{6}(3) = \frac{1}{2}
\]

Since both sides are equal (\(\frac{1}{2} = \frac{1}{2}\)), our solution is confirmed. The number is indeed \( \boxed{6} \).