Sum & Difference

1. 1Bracket the SumTogether, the two numbers make the known total, so a + b = 10.a + b = 10
2. 2Identify the DifferenceBecause a is 4 greater than b, the amount beyond b is exactly the difference: a − b = 4.a − b = 4
3. 3Isolate the Extra PieceThe extra amount is 4. The camera leans in on it — this is exactly the piece we are about to peel off the longer bar.Extra piece = 4
4. 4Peel the Extra Off FirstPeel the extra off first: the difference piece of 4, bracket and all, parks below. After subtracting the difference from a, both numbers are equal: a − 4 = b.a − 4 = b
5. 5Compare Remaining BarsThe two equal numbers now add to 6, the original total minus the difference.Part A = Part B
6. 6Solve for Smaller Quantity (b)The number 6 just grew out of the sum-minus-difference label on the diagram itself. Split the remaining total, 6, equally between the two numbers — each half is the smaller number b.2b = 6 ➔ b = 3
7. 7Solve for Larger Quantity (a)Now fly the extra piece back and land it on a: the larger number is 4 more than the smaller, so a = 3 + 4 = 7.a = 3 + 4 = 7
8. 8Check the AnswerNever trust an answer before testing it against BOTH givens: larger + smaller must equal the sum 10, and larger − smaller must equal the difference 4.7 + 3 = 10 ✓ · 7 − 3 = 4 ✓
9. 9The General Reusable RuleFor any two numbers, subtract the difference from the sum and halve it to get the smaller; add them and halve to get the larger. Before a new problem, classify it: given the sum and the difference → peel the extra. What if you are given the sum and a ratio instead?smaller = 3, larger = 7

Change the numbers to test the engine with any sum & difference problem: