The numerator of a fraction is 4 less than its denominator. If the numerator…
2024
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the new denominator becomes 8 times the new numerator. Find the fraction.
Answer: A. \(\frac{3}{7}\) — Concept For a fraction with numerator n and denominator d, a verbal difference gives an equation between n and d. When both parts are changed, apply those…
- A.
\(\frac{3}{7}\)
- B.
\(\frac{3}{9}\)
- C.
\(\frac{3}{8}\)
- D.
\(\frac{2}{7}\)
Show answer & explanation
Correct answer: A
Concept
For a fraction with numerator n and denominator d, a verbal difference gives an equation between n and d. When both parts are changed, apply those changes to the same variables before forming the second equation.
Application
Let the numerator be n. Since it is 4 less than the denominator, d = n + 4.
After the stated changes, the new numerator is n − 2 and the new denominator is d + 1 = n + 5.
The new denominator is 8 times the new numerator, so n + 5 = 8(n − 2).
Expanding gives n + 5 = 8n − 16.
Collecting like terms gives 21 = 7n, hence n = 3.
Then d = n + 4 = 3 + 4 = 7.
Cross-check
For 3/7, the numerator–denominator difference is 7 − 3 = 4.
After the changes, the values are 3 − 2 = 1 and 7 + 1 = 8; indeed, 8 = 8 × 1.
Result
Therefore, the required fraction is 3/7.