Simplify, scale, and solve for the missing value
Simplify, scale, and solve for the missing value
A ratio compares two quantities. Written 16:24, it says that for every 16 of the first there are 24 of the second. Ratios are scale-free — 16:24, 8:12, and 2:3 all describe exactly the same relationship.
To simplify, divide both terms by their greatest common divisor. For 16 and 24 the GCD is 8, giving 2:3. This is the ratio in lowest terms, and it is the form that makes the relationship easiest to see.
A proportion states that two ratios are equal. Given three values, the fourth follows:
a : b = c : d → a × d = b × c → d = (b × c) ÷ a
If 16:24 = 40:d, then d = (24 × 40) ÷ 16 = 60. This cross-multiplication method underlies unit conversion, recipe scaling, map reading, and most everyday proportion problems.
Worked example: a recipe for 4 people needs 300 g of flour. For 7 people, 4:300 = 7:d, so d = (300 × 7) ÷ 4 = 525 g.
These express related but distinct things, and conflating them causes real errors.
The ratio 2:3 compares two parts to each other. The total is 5 parts, so the first quantity is 2/5 of the whole, or 40%.
The frequent mistake is reading 2:3 as “two thirds”. It is not — 2:3 means 2/5 and 3/5 of the total. Two thirds would be the ratio 2:1.
To convert a ratio to a percentage of the whole: divide each term by the sum of all terms. For 2:3, that is 2/5 = 40% and 3/5 = 60%.
Ratios extend naturally. A concrete mix of 1:2:4 means one part cement, two parts sand, four parts aggregate — seven parts total.
To split a quantity by a ratio, divide by the total number of parts and multiply. Splitting $2,100 in the ratio 2:3:5 means 10 parts total, so each part is $210, giving $420, $630, and $1,050. The parts always sum back to the original.
To simplify a multi-term ratio, find the GCD of all terms. For 12:18:30, the GCD is 6, giving 2:3:5.
Aspect ratios. 16:9 describes screen proportions independent of resolution — 1920×1080 and 3840×2160 are both 16:9.
Scale drawings and maps. A 1:50 architectural drawing means 1 cm on paper represents 50 cm in reality. A 1:25,000 map means 1 cm is 250 metres.
Mixing. Paint, fuel, concrete, and chemical solutions are all specified by ratio because the proportions matter and the total quantity does not.
Finance. Debt-to-income, price-to-earnings, and current ratio are all comparisons of two figures where the relationship is more informative than either number alone.
Photography and design. The golden ratio, roughly 1:1.618, and the rule of thirds both structure composition proportionally rather than by fixed measurement.
Divide both terms by their greatest common divisor. For 16:24 the GCD is 8, giving 2:3. The simplified ratio expresses the same relationship in the smallest whole numbers.
No. In the ratio 2:3 there are 5 parts total, so the first quantity is 2/5 of the whole, or 40%. Two thirds would be the ratio 2:1.
Cross-multiply. If a:b equals c:d, then a times d equals b times c, so any missing term can be found by rearranging.
Add the terms to get the total number of parts, divide the amount by that total to find the value of one part, then multiply by each term.