Question

Mrs. X spends Rs. 535 in purchasing some shirts and ties for her husband. If shirts cost Rs. 43 each and the ties cost Rs. 21 each, then what is the ratio of the shirts to the ties, that are purchased?

• A. 1 : 2
• B. 2 : 1
• C. 2 : 3
• D. 3 : 4

Hint:

Write the total cost as 43x + 21y = 535 and find the smallest whole-number solution, or test each answer ratio directly against the total.

Answer 1

The Correct Option is B

Step 1: Set up the equation.
Let the number of shirts purchased be \(x\) and the number of ties be \(y\). Since each shirt costs Rs. 43 and each tie costs Rs. 21, and the total spent is Rs. 535:
\[ 43x + 21y = 535 \]
Both \(x\) and \(y\) must be non-negative whole numbers, since you cannot buy a fraction of a shirt or tie.

Step 2: Solve for whole number values.
Taking the equation mod 21, and noting \(43 \equiv 1 \pmod{21}\), we get \(x \equiv 535 \pmod{21}\). Now \(535 = 21(25) + 10\), so \(x \equiv 10 \pmod{21}\).
Because \(43x \le 535\), we need \(x \le 12.4\), so the only value of \(x\) in this range congruent to 10 mod 21 is \(x = 10\).
Substituting back: \(21y = 535 - 430 = 105\), so \(y = 5\).

Final Answer:
The ratio of shirts to ties is \(10 : 5 = 2 : 1\). Options like 1:2 or 2:3 don't satisfy the cost equation with whole numbers of shirts and ties, only \(x=10, y=5\) does. \[ \boxed{\text{Shirts : Ties} = 2 : 1} \]