A donation box can receive only cheques of ₹100, ₹250 and ₹1000. On the last day of the month, the box contained exactly 90 cheques totalling ₹42600. If the number of ₹1000 cheques was the maximum possible, what is the sum of the numbers of ₹100 and ₹250 cheques?
Answer and solution
Correct answer: 53
Solution
Let the counts be $x$ (₹100), $y$ (₹250), $z$ (₹1000).
$x + y + z = 90$ and $100x + 250y + 1000z = 42600 \Rightarrow 2x + 5y + 20z = 852$
Subtract $2(x + y + z) = 180$: $3y + 18z = 672 \Rightarrow y = 224 - 6z$
$x = 90 - y - z = 5z - 134$
$x \ge 0 \Rightarrow z \ge 27$; $y \ge 0 \Rightarrow z \le 37$. Maximum $z = 37$
$x = 51,\; y = 2 \Rightarrow x + y = 53$
$x + y + z = 90$ and $100x + 250y + 1000z = 42600 \Rightarrow 2x + 5y + 20z = 852$
Subtract $2(x + y + z) = 180$: $3y + 18z = 672 \Rightarrow y = 224 - 6z$
$x = 90 - y - z = 5z - 134$
$x \ge 0 \Rightarrow z \ge 27$; $y \ge 0 \Rightarrow z \le 37$. Maximum $z = 37$
$x = 51,\; y = 2 \Rightarrow x + y = 53$