A jug holds 5 liters and a mug holds 3 liters. A bottle holds 29 liters. How many ways can the bottle be filled exactly using jugs and mugs?

hard 3 marks Class 3 · Mathematics · Jugs and Mugs

Answer

2 ways

The correct answer is 2 ways — These are the only integer solutions, so there are 2 ways.

Step-by-step solution

Let $j$ be the number of jugs and $m$ the number of mugs.
Set up the equation for the total volume: $5 j + 3 m = 29$ .
Express $j$ in terms of $m$ : $j = \frac{29 - 3 m}{5}$ .
For $j$ to be an integer, $29 - 3 m$ must be divisible by 5.
Solve the congruence $3 m e q u i v 29 (mod 5)$ , i.e. $3 m e q u i v 4 (mod 5)$ .
Multiply both sides by the inverse of 3 modulo 5, which is 2: $m e q u i v 8 e q u i v 3 (mod 5)$ .
Thus $m = 3 + 5 k$ for some non‑negative integer $k$ .
The condition $m l e 9$ (since $3 m l e 29$ ) restricts $k$ to $0$ or $1$ .
For $m = 3$ , $j = \frac{29 - 9}{5} = 4$ .
For $m = 8$ , $j = \frac{29 - 24}{5} = 1$ .
These are the only integer solutions, so there are 2 ways.

Common mistake: Ignoring modulus conditions or negative solutions

Concept tested: Linear Diophantine equations