Difference & Ratio

1. 1The Parts LandBar A has 5 equal parts and Bar B has 2 equal parts, landing one by one — every part the same size as every other.a : b = 5 : 2
2. 2Lay B Over ASlide B's row up to lie exactly over the first 2 parts of A. The matched parts dim — matched parts make no difference; only the overhang is the difference.5 parts − 2 matched parts
3. 3The Overhang Is the DifferenceThe overhang is revealed: count its parts one by one — 3 parts, worth exactly the difference of 30. Common mistake: dividing the difference by the total parts of both bars — wrong, because only the overhang carries the value of the difference.5 − 2 = 3 parts = 30
4. 4Find the Value of One Part FirstFind the value of one part first: if 3 equal units have a value of 30, one unit is 30 ÷ 3 = 10.1 part = 30 ÷ 3 = 10
5. 5Solve for Quantity ASweep A's 5 parts while the counter adds one part-value at a time: A is 50.a = 5 × 10 = 50
6. 6Solve for Quantity BSweep B's 2 parts the same way: B is 20.b = 2 × 10 = 20
7. 7Check on the DiagramLay B over A once more: the overhang must come back as exactly the given difference. 50 − 20 = 30 ✓.50 − 20 = 30 ✓
8. 8Summary & DiagnosisGiven the SUM → add the parts. Given the DIFFERENCE → subtract the parts. Facing a new problem, the first question is: what are you given?a = 50, b = 20

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