Grades45
Finding two numbers is a foundation family of Grade 4 and Grade 5 thinking maths: the problem never states the two numbers directly, only how they relate — a sum and a difference, a sum and a ratio, a difference and a ratio — or, in age problems, how that relation shifts over time. Every member of the family is unlocked by the same tool, the bar model: once the bars are drawn, it becomes visible whether the sum, the difference or the ratio must be split, and into how many parts. This page gathers all four lessons of the family, with fully worked examples, so parents can coach at home.
Spot this family when two unknown quantities are tied together by a pair of facts: a total plus how much one exceeds the other (sum & difference); a total plus how many times — or what fraction — one number is of the other (sum & ratio); a difference together with a ratio (difference & ratio); or the ages of two people at different times (age problems, where the unchanging age gap is the key). Two unknowns plus one such pair of relations means a finding-two-numbers problem.
Two numbers have a sum of 12 and a difference of 4. What is the larger number?
Key idea: Smaller first: (sum − difference) ÷ 2, then add the difference back.
You know the sum (12) and the difference (4) of two numbers — a sum–difference problem.
Remove the difference so the sum splits into two equal parts.
Smaller = (12 − 4) ÷ 2 = 4.
Larger = 4 + 4 = 8.
So: So the larger number is 8.
Two quantities are in the ratio 2:3. Their total is 25. What is the smaller quantity?
Key idea: Divide the total by the total number of parts to find one part.
So: So the smaller quantity is 10.
Two numbers are in the ratio 3:1. Their difference is 10. What is the larger number?
Key idea: The difference is worth (a − b) parts — divide it to find one part.
So: So the larger number is 15.
In 5 years, a mother and her son will be 50 years old in total. The mother is 30 years older than the son. How old is the son now?
Key idea: The age gap never changes — solve in the future year, then step back.
So: So the son is 5 years old now.
Two numbers have a total of 10. One number is 4 greater than the other. What are the two numbers?
Shortcuts: Space (play), Arrow keys (step), R (reset), F (fullscreen)
The two unknown numbers add to 10, and the larger is 4 greater than the smaller. We need to find both values.
The big question is hanging on the stage — the whole lesson will answer it.
Change the numbers and watch the diagram redraw itself:
The family is core Grade 4 material and continues into Grade 5 through age problems. All four types appear regularly in olympiad papers and selective Grade 6 entrance exams, often hidden inside perimeter and age contexts.
With the bar model. Drawing the two bars exactly as the problem states the relation — levelling the longer bar, or splitting it into ratio parts — shows which quantity must be divided and by how many parts, instead of guessing a formula.
In exactly one step — finding the value of one part. Sum & ratio divides the total by the total number of parts; difference & ratio divides the difference by the gap in parts. That single step is the most common mix-up in the whole family.
