Two numbers have a sum of 84. Which of the following ratios can they not be in: 5 : 7, 13 : 8, 1 : 3 or 3 : 2?

Introduction / Context:
This question checks whether a given ratio is compatible with a specified sum of two numbers. Such problems involve expressing the numbers in terms of a ratio and verifying whether the sum matches the required total with integer values. This is important in ratio and proportion topics where sums and ratios are linked.

Given Data / Assumptions:

• The sum of two numbers is 84.
• Possible ratios to test: 5 : 7, 13 : 8, 1 : 3, 3 : 2.
• We must determine which ratio the two numbers cannot have if they are to sum to 84 as whole numbers.

Concept / Approach:
For each ratio m : n, we can represent the two numbers as m*k and n*k for some constant k. The sum is then (m + n)k. For the sum to be exactly 84, k must be 84 / (m + n). To get integer numbers, k must be an integer. Therefore, we check for each ratio whether 84 is divisible by (m + n). If it is not divisible, that ratio is not possible for two whole numbers summing to 84.

Step-by-Step Solution:

Verification / Alternative check:
We can explicitly compute pairs from the valid ratios and check sums. For 5 : 7, 35 + 49 = 84; for 13 : 8, 52 + 32 = 84; for 1 : 3, 21 + 63 = 84. In contrast, attempting 3 : 2 gives numbers 3k and 2k with sum 5k = 84, leading to k = 84/5 = 16.8, so one or both numbers would not be integers. Hence 3 : 2 is the only invalid ratio if we require integer values.

Why Other Options Are Wrong:

• 5 : 7, 13 : 8 and 1 : 3 all yield integer-valued pairs whose sum is 84, so they are possible.
• They therefore cannot be the “not possible” ratio asked by the question.

Common Pitfalls:
Some students may assume that any ratio can work without checking the divisibility of 84 by the sum of ratio parts. Others may make arithmetic mistakes when dividing 84, especially if done mentally. Always compute (m + n) and then see if 84 divides evenly to ensure integer solutions.

Final Answer:
The numbers cannot be in the ratio 3 : 2 if their sum is 84.