If every 2 out of 3 readymade shirts need alterations in the collar, every 3 out of 4 need alterations in the sleeves, and every 4 out of 5 need it in the body, how many alterations will be required for 60 shirts?

### Concept & Formula The problem requires computing the sum total of individual defects across three mutually exclusive components (collar, sleeves, body) for a given batch of items. $$ \text{Total Alterations} = \text{Collar Alterations} + \text{Sleeves Alterations} + \text{Body Alterations} $$ ### Step-by-Step Solution * **Given:** * Total shirts = $60$ * Collar alteration rate = $\frac{2}{3}$ * Sleeves alteration rate = $\frac{3}{4}$ * Body alteration rate = $\frac{4}{5}$ * **Calculation:** * Alterations needed in the collar: $$ \frac{2}{3} \times 60 = 40 $$ * Alterations needed in the sleeves: $$ \frac{3}{4} \times 60 = 45 $$ * Alterations needed in the body: $$ \frac{4}{5} \times 60 = 48 $$ * Total alterations required: $$ \text{Total} = 40 + 45 + 48 = 133 $$ ### Exam Strategy & Shortcut Factor out the total volume of shirts to combine the fractional operations into a single step: $$ \text{Total} = 60 \times \left( \frac{2}{3} + \frac{3}{4} + \frac{4}{5} \right) $$ $$ \text{Total} = 60 \times \frac{2}{3} + 60 \times \frac{3}{4} + 60 \times \frac{4}{5} = 40 + 45 + 48 = 133 $$ Since 60 is perfectly divisible by the denominators 3, 4, and 5, evaluating each term independently is exceptionally swift. ### Common Pitfall A common misinterpretation is treating the fractions dependently (e.g., computing alterations on the remaining shirts left after the previous alteration). The question states "every 2 out of 3 readymade shirts...", meaning these proportions apply to the entire initial pool of 60 shirts independently. ### Final Answer **Therefore, the correct answer is 133.**