Question:
The numerator and denominator of a fraction is in the ratio 2:3. If 6 are subtracted from the numerator the value of the fraction becomes 2/3 of the original fraction. The numerator of the original fraction is,
The numerator and denominator of a fraction is in the ratio 2:3. If 6 are subtracted from the numerator the value of the fraction becomes 2/3 of the original fraction. The numerator of the original fraction is,
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Let numerator = 2x and denominator = 3x, then set (2x-6)/3x equal to two-thirds of 2/3 and solve for x.
Updated On: Jul 15, 2026
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The Correct Option is C
Solution and Explanation
Step 1: Represent the fraction using the given ratio.
The numerator and denominator are in the ratio 2:3, so let the numerator be 2x and the denominator be 3x, for some positive number x. The original fraction is then 2x / 3x, which simplifies to 2/3.
Step 2: Write the new fraction after subtracting 6 from the numerator.
Subtracting 6 from the numerator only, the denominator stays 3x, gives the new fraction (2x - 6) / 3x.
Step 3: Use the given condition relating the new fraction to the original.
The new fraction equals 2/3 of the original fraction. Since the original fraction is 2/3, two-thirds of it is (2/3) x (2/3) = 4/9. So: (2x - 6) / 3x = 4/9.
Step 4: Cross-multiply and solve for x.
9(2x - 6) = 4(3x). Expanding both sides: 18x - 54 = 12x. Subtracting 12x from both sides: 6x - 54 = 0, so 6x = 54, giving x = 9.
Step 5: Find the original numerator.
The numerator was defined as 2x, so numerator = 2 x 9 = 18.
Step 6: Verify the answer.
With x = 9, numerator = 18, denominator = 27, so the original fraction is 18/27 = 2/3, matching the given ratio. Subtracting 6 from the numerator gives 12/27 = 4/9. Checking, (2/3) of the original fraction (2/3) is (2/3) x (2/3) = 4/9, which matches exactly. So numerator = 18 is confirmed, matching option (3).
The numerator and denominator are in the ratio 2:3, so let the numerator be 2x and the denominator be 3x, for some positive number x. The original fraction is then 2x / 3x, which simplifies to 2/3.
Step 2: Write the new fraction after subtracting 6 from the numerator.
Subtracting 6 from the numerator only, the denominator stays 3x, gives the new fraction (2x - 6) / 3x.
Step 3: Use the given condition relating the new fraction to the original.
The new fraction equals 2/3 of the original fraction. Since the original fraction is 2/3, two-thirds of it is (2/3) x (2/3) = 4/9. So: (2x - 6) / 3x = 4/9.
Step 4: Cross-multiply and solve for x.
9(2x - 6) = 4(3x). Expanding both sides: 18x - 54 = 12x. Subtracting 12x from both sides: 6x - 54 = 0, so 6x = 54, giving x = 9.
Step 5: Find the original numerator.
The numerator was defined as 2x, so numerator = 2 x 9 = 18.
Step 6: Verify the answer.
With x = 9, numerator = 18, denominator = 27, so the original fraction is 18/27 = 2/3, matching the given ratio. Subtracting 6 from the numerator gives 12/27 = 4/9. Checking, (2/3) of the original fraction (2/3) is (2/3) x (2/3) = 4/9, which matches exactly. So numerator = 18 is confirmed, matching option (3).
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