Ratio Calculator

What's the key thing to understand about Simplifying Ratios?

A ratio compares two or more quantities, and simplifying it means writing it in its lowest terms — the smallest whole numbers that preserve the same proportion. To simplify, divide every value by their Greatest Common Divisor (GCD), which is the largest number that divides all of them. Worked example: 12:8. The GCD of 12 and 8 is 4, so dividing both by 4 gives 3:2 — the simplest form. Another: 100:75. The GCD is 25, giving 4:3. If you can't immediately spot the GCD, simplify in steps: 24:36 → divide by 2 → 12:18 → divide by 2 → 6:9 → divide by 3 → 2:3. You arrive at the same answer either way. A ratio is in lowest terms when its values share no common factor other than 1 (they are 'coprime'). For three-part ratios like 6:9:12, find the GCD of all three (here, 3) and divide each — giving 2:3:4. Simplified ratios matter because they make comparisons and scaling easier — 3:2 is instantly clearer than 75:50, even though they're equal. They also reveal underlying relationships that bigger numbers can obscure: 1750:1250 looks complex until you simplify it to 7:5. Always express final ratio answers in simplest form unless a question specifically asks otherwise.

What do I need to know about Scaling Proportions?

Once you understand a ratio, the real power is scaling it up or down to whatever quantity you need, while keeping the relationship intact. Worked example: if paint is mixed at 2 parts blue to 3 parts white, and you want to use 10 litres of blue, you need 3/2 × 10 = 15 litres of white — so 25 litres of paint total in the same 2:3 proportion. The general method is to find the scaling factor: how much you multiplied (or divided) one part by, then apply the same factor to the others. If a recipe for 4 people uses 200 g of pasta, 300 g of sauce, and 50 g of cheese (a 4:6:1 ratio in some sense, or 200:300:50 by weight), scaling to 6 people means multiplying every quantity by 6/4 = 1.5: 300 g pasta, 450 g sauce, 75 g cheese. The ratio between the ingredients stays exactly the same; only the absolute amounts change. Map and model scales work the same way: 1:25,000 means 1 cm on the map represents 25,000 cm (250 m) in reality, so a route 4 cm long on the map is 1 km in reality. A 1:72 model car is 72 times smaller than the real thing in every dimension. Whenever you scale, every quantity in the ratio must be multiplied (or divided) by the same factor — scaling one without the others breaks the proportion and the whole point of having a ratio.

What's the key thing to understand about Common Uses?

Ratios appear so often in everyday life that recognising them as ratios makes problems easier to solve. Cooking and baking rely on them: a 1:1 ratio of flour and butter (by weight) is the classic shortcrust pastry; a 1:2 hot water to dry rice ratio cooks plain rice; bread flour to water at ~5:3 by weight gives standard 60% hydration dough. Building and DIY: a standard concrete mix is 1:2:4 cement to sand to aggregate by volume; mortar is typically 1:3 or 1:4 cement to soft sand; lime mortar varies for different jobs. Map scales (1:25,000 for OS Explorer maps, 1:50,000 for OS Landranger), model railways (00 gauge is 1:76.2, N gauge 1:148), and architectural drawings all use ratios to relate the drawn or modelled object to reality. Photography: aspect ratios describe image proportions (4:3, 16:9, 3:2). Finance: gearing, debt-to-equity, and price-to-earnings ratios let analysts compare companies of different sizes on equal footing. Chemistry: stoichiometric ratios in equations show how much of each reactant produces how much product. Even gambling odds are ratios in disguise (5:1 odds mean five losses expected for every win, implying a 1/6 probability). Recognising when a problem is fundamentally about ratios — same proportion, different quantities — often makes the solution obvious.

What's the key thing to understand about Exam Tips and Common Errors?

A few habits prevent the most common ratio mistakes. First, when scaling, always apply the same factor to every part of the ratio — multiplying one and forgetting the other breaks the proportion. Second, when sharing in a ratio (e.g. 'split £180 in the ratio 2:3:4'), add the parts to get the total parts (2+3+4 = 9), divide the total by that (£180 ÷ 9 = £20 per part), then multiply by each ratio number (£40 : £60 : £80). The classic mistake is treating the ratio numbers as percentages, which only works by accident. Third, ratios are unitless when comparing the same kind of quantity, but the units must match before you take the ratio — comparing 30 cm to 2 m means converting both to the same unit first (30:200, simplified to 3:20), not 30:2. Fourth, a fraction and a ratio can express the same idea differently: '2 parts blue out of every 5 parts paint' is equivalent to a 2:3 blue-to-white ratio, but they're not the same number — be clear whether you're working in 'parts of' or 'compared to'. Fifth, when comparing ratios for equivalence (is 6:9 the same as 8:12?), the test is whether the cross-multiplications match: 6 × 12 = 72, 9 × 8 = 72, so yes, they're equal. Finally, always simplify your final answer. This calculator handles the arithmetic; understanding what ratios represent makes word problems much easier to set up correctly.

Worked Example: Dividing a Quantity in a Given Ratio

A common ratio task is splitting a total amount into parts that follow a given ratio, rather than simplifying or scaling an existing pair of numbers. Suppose three business partners agree to split a $720 profit in the ratio 3:4:5 based on their investment shares. First add the parts of the ratio: 3 + 4 + 5 = 12 total parts. Divide the total amount by the number of parts to find the value of one part: $720 ÷ 12 = $60 per part. Now multiply each ratio term by this unit value: Partner A gets 3 × $60 = $180, Partner B gets 4 × $60 = $240, and Partner C gets 5 × $60 = $300. As a check, add the three shares back together: $180 + $240 + $300 = $720, which matches the original total, confirming the split is correct. This "total ÷ parts × each term" method works for any number of ratio terms and any total, and it's the same logic used behind the scenes whenever a ratio calculator splits a quantity rather than just simplifying it.

Ratios, Fractions, and Percentages Compared

A two-term ratio a:b can always be rewritten as a fraction of the whole, a / (a + b), and then as a percentage. Seeing the three forms side by side makes it easier to translate a ratio into a share of a total.

To build this table for any ratio, add the two parts to get the whole, then put the first term over that sum. For 5:7, the whole is 5 + 7 = 12, so the fraction is 5/12, which as a decimal is 0.4167, or 41.67%. This conversion is especially useful in exam questions that mix ratio and percentage language, such as "a mixture is made in the ratio 2:3 — what percentage of the mixture is the first ingredient?" The answer is simply 2/(2+3) = 40%.

Using Ratios as Rates: Speed, Density, and Unit Price

A ratio doesn't have to compare two amounts of the same thing — it can also compare two different quantities, in which case it becomes a rate. Three everyday examples show how this works. Speed as a ratio: a car covers 180 km in 3 hours. Written as a ratio of distance to time, that's 180:3, which simplifies by dividing both sides by 3 to 60:1 — in other words, 60 km per 1 hour, or 60 km/h. Density as a ratio: a sample has a mass of 270 g and a volume of 30 cm³. The ratio 270:30 simplifies (dividing by 30) to 9:1, meaning 9 grams for every 1 cm³, so the density is 9 g/cm³. Unit price as a ratio: five apples cost $2.50 in total. The ratio of cost to quantity is 2.50:5, which simplifies (dividing by 5) to 0.50:1, giving a unit price of $0.50 per apple. In every case, simplifying the ratio down to a denominator of 1 is exactly what turns a raw ratio into a familiar "per unit" rate — km per hour, grams per cm³, or dollars per item.