A farmer divides his herd of cows among his four sons so that first son gets one-half of the herd, the second son gets one-fourth, the third son one-fifth and the fourth son 7 cows. The total number of cows in the herd is

## Concept & Logic This problem involves fractional parts of a whole. The sum of the fractions received by the first three sons, plus the absolute number received by the fourth son, must equal the total herd size. $$ \frac{x}{2} + \frac{x}{4} + \frac{x}{5} + Remainder = x $$ ## Step-by-Step Solution * **Given:** First son gets $\frac{1}{2}$, second gets $\frac{1}{4}$, third gets $\frac{1}{5}$, and fourth gets 7 cows. Let total cows = $x$. * **Calculation:** Set up the primary equation based on the total: $\frac{x}{2} + \frac{x}{4} + \frac{x}{5} + 7 = x$ * **Deduction:** Find the Lowest Common Multiple (LCM) of the denominators 2, 4, and 5, which is 20. * **Calculation:** Rewrite the fractions with the common denominator: $\frac{10x}{20} + \frac{5x}{20} + \frac{4x}{20} + 7 = x$ $\frac{19x}{20} + 7 = x$ * **Calculation:** Isolate $x$: $7 = x - \frac{19x}{20}$ $7 = \frac{x}{20}$ $x = 7 \times 20 = 140$ ## Exam Strategy & Shortcut Instead of algebra, use LCM directly on the fractions. The fractions are $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{1}{5}$. The LCM of denominators is 20. Assume the total herd is 20 units. First son gets 10 units. Second gets 5 units. Third gets 4 units. Total given out = $10 + 5 + 4 = 19$ units. Remaining = 1 unit. The problem states the remainder is 7 cows. Therefore, 1 unit = 7 cows. Total herd = 20 units $\times 7 = 140$ cows. ## Common Pitfall A frequent mistake is calculating fractions of the *remainder* instead of fractions of the *total herd*. Read the wording carefully: "one-half of the herd, one-fourth..." implies all fractions apply to the original total, not consecutive remainders. ## Final Answer Therefore, the correct answer is **140**.