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:
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One quarter of the difference between 6 and 4: \[ \frac{1}{4}(6 - 4) = \frac{1}{4}(2) = \frac{1}{2} \]
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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 \( \boxed{6} \).
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