Ratio Calculator

About Ratio Calculator

How Does Cross Multiplication Work?

A proportion states that two ratios are equal: A:B = C:D. If three of the four values are known, the missing one can be found using cross multiplication. The method works because equal fractions have equal cross products.

A/B = C/D means A x D = B x C

Worked example: If 3:5 = 12:?, find the missing value.

1. Set up the proportion: 3/5 = 12/?
2. Cross multiply: 3 x ? = 5 x 12
3. 3 x ? = 60
4. ? = 60 / 3 = 20
5. Answer: 3:5 = 12:20

Cross multiplication is sometimes called the "cross your heart" method in classrooms because the lines drawn between the numerators and denominators resemble a heart outline. The technique dates back to basic algebra and is covered in most curricula from around age 11-12, though understanding why it works (equivalent fractions) matters more than memorising the procedure.

Second example (real-world): A map has a scale of 1:25,000. A distance measures 8.4 cm on the map. How far is that in real life?

1. 1/25,000 = 8.4/?
2. ? = 25,000 x 8.4 = 210,000 cm
3. Convert: 210,000 cm = 2,100 m = 2.1 km

How to Simplify a Ratio

To simplify a ratio, divide both parts by their greatest common factor (GCF). The simplified ratio expresses the same proportion using the smallest possible whole numbers.

Original Ratio	GCF	Simplified
12:18	6	2:3
45:60	15	3:4
24:36:48	12	2:3:4
100:250	50	2:5
0.5:1.5	-	1:3
2.4:3.6	-	2:3

For decimal ratios, the calculator first converts to whole numbers by multiplying both sides by the appropriate power of 10, then finds the GCF. For example, 0.5:1.5 becomes 5:15 after multiplying by 10, then divides by GCF 5 to get 1:3. For finding the GCF directly, see the GCF and LCM calculator.

A common mistake when simplifying is dividing by a common factor that is not the greatest common factor. Dividing 24:36 by 2 gives 12:18, which is simpler but not fully simplified. Always check whether the result can be reduced further, or find the GCF in one step.

How to Scale a Ratio

Scaling multiplies both parts of a ratio by the same factor. The proportion stays the same but the quantities change - useful for adjusting recipe quantities, mixing solutions, or resizing designs.

Worked example: A recipe uses flour and sugar in a 3:2 ratio. You need 750 g of flour. How much sugar?

1. Scale factor: 750 / 3 = 250
2. Sugar: 2 x 250 = 500 g
3. Scaled ratio: 750:500 (still 3:2)

This is the same principle used by the recipe scaler, which handles full ingredient lists at once. For a single ratio scaling, this calculator shows the working.

Second example (concrete mixing): A general-purpose concrete mix uses a 1:2:3 ratio of cement to sand to gravel. For a project needing 6 bags of cement:

1. Scale factor: 6 / 1 = 6
2. Sand: 2 x 6 = 12 parts
3. Gravel: 3 x 6 = 18 parts
4. Scaled ratio: 6:12:18 (still 1:2:3)

The 1:2:3 ratio is the standard general-purpose concrete mix, producing roughly 3,000 psi compressive strength. High-strength applications like columns and beams use a richer 1:1.5:3 mix, while slabs and floors commonly use a leaner 1:2:4 ratio.

Ratios, Fractions, and Percentages

Ratios, fractions, and percentages are all ways of expressing the same relationship. Converting between them is a fundamental maths skill:

Ratio	Fraction (of total)	Percentage	Decimal
1:1	1/2 each	50% each	0.5 each
1:2	1/3 and 2/3	33.3% and 66.7%	0.333 and 0.667
1:3	1/4 and 3/4	25% and 75%	0.25 and 0.75
2:3	2/5 and 3/5	40% and 60%	0.4 and 0.6
3:4	3/7 and 4/7	42.9% and 57.1%	0.429 and 0.571
1:1:1	1/3 each	33.3% each	0.333 each

To convert a ratio A:B to a percentage, divide each part by the total (A + B), then multiply by 100. For example, 3:7 means 3/(3+7) = 30% and 7/(3+7) = 70%. The percentage calculator handles these conversions directly.

Where Are Ratios Used in Real Life?

Field	Example	Typical Ratio	Notes
Cooking	Rice to water	1:2 (white), 1:1.5 (brown)	Most grain-to-water ratios follow fixed proportions
Construction	Cement:sand:gravel	1:2:3 general purpose	Produces approx. 3,000 psi concrete
Photography	Image aspect ratios	4:3, 16:9, 3:2, 1:1	3:2 comes from 35mm film dimensions
Maps	Scale	1:25,000 or 1:50,000	Ordnance Survey Explorer maps use 1:25,000
Finance	Debt-to-income	Under 36% (9:25) is healthy	Above 43% usually disqualifies for a mortgage
Chemistry	Molar ratios	2H2 + O2 gives 2H2O	Stoichiometry is ratio maths applied to molecules
Architecture	Golden ratio	1:1.618	Used since ancient Greece; Le Corbusier's Modulor system
Screens	Display ratios	16:9, 21:9	16:9 became the HDTV standard in the 1990s

In finance, the debt-to-income (DTI) ratio divides monthly debt payments by gross monthly income. Lenders generally consider a DTI under 36% healthy, and most mortgage lenders set 43% as the maximum qualifying threshold. The DTI calculator handles this specific ratio.

The Golden Ratio - 1:1.618

The golden ratio (phi, approximately 1.6180339887) is probably the most famous ratio in mathematics. Two quantities are in the golden ratio if their ratio equals the ratio of their sum to the larger quantity: a/b = (a + b)/a = 1.618...

The ratio appears throughout nature - in the spiral arrangement of sunflower seeds, the proportions of nautilus shells, and the branching patterns of trees. In design, architects have used it since ancient Greece. The Parthenon's facade proportions approximate the golden ratio, and Swiss architect Le Corbusier built his entire Modulor measurement system around it, applying it to projects like the housing complex La Cite Radieuse in Marseille. The golden ratio calculator explores these proportions in more depth.

The golden ratio is also closely tied to the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21...). As the sequence progresses, the ratio between consecutive numbers converges on 1.618 - for instance, 21/13 = 1.615 and 34/21 = 1.619.

Common Aspect Ratios in Detail

Screen and image ratios are among the most commonly looked-up ratios. The original 4:3 (1.33:1) format dates back to the 1890s, when William Dickson used standard Eastman Kodak 35mm film in Thomas Edison's lab. The usable image area on four-perforation 35mm frames worked out to roughly 1.33:1, and early television adopted the same ratio when NTSC set it as the broadcast standard in 1941.

In the 1950s, cinema moved to widescreen formats to differentiate from TV. 20th Century Fox introduced CinemaScope in 1953 with the film The Robe, using anamorphic lenses to achieve a 2.35:1 ratio. The modern 16:9 (1.78:1) standard was proposed by Kerns H. Powers of the SMPTE Working Group in 1984. Engineers chose it because 1.78 is the geometric mean between 4:3 (1.33) and the typical cinema ratio of 2.35:1, making it a practical compromise for displaying both TV and film content on one screen. The aspect ratio calculator helps with resizing images and video to these standard ratios.

Ratio	Decimal	Where Used
1:1	1.000	Instagram square, profile photos
4:3	1.333	Classic TV (NTSC 1941), iPad, 1024x768
3:2	1.500	35mm film, many DSLR photos
16:10	1.600	MacBook displays, some monitors
16:9	1.778	HD/4K TV, YouTube, 1920x1080
21:9	2.333	Ultrawide monitors, CinemaScope
9:16	0.563	Phone vertical video, TikTok, Stories

Comparing Two Ratios

Two ratios are equivalent (proportional) if they simplify to the same ratio. The compare mode converts each to a decimal to check:

Example: Are 15:25 and 21:35 equivalent?

• 15:25 = 15/25 = 0.6, simplifies to 3:5
• 21:35 = 21/35 = 0.6, simplifies to 3:5
• Both simplify to 3:5, so they are equivalent

Example: Are 4:7 and 12:20 equivalent?

• 4:7 = 4/7 = 0.5714...
• 12:20 = 12/20 = 0.6
• Different decimals, so they are not equivalent (4:7 vs 3:5)

Comparing ratios comes up in quality control, recipe adjustments, and verifying that scaled quantities maintain the correct proportions. In chemistry, comparing molar ratios determines whether a reaction has the right stoichiometric balance.

Common Mistakes with Ratios

A few errors come up repeatedly in ratio problems:

• Confusing part-to-part with part-to-whole: A 2:3 ratio of boys to girls means 2 out of every 5 students are boys (40%), not 2 out of 3 (67%). The total is 2 + 3 = 5.
• Adding instead of multiplying when scaling: Scaling 2:3 by a factor of 4 gives 8:12 (multiply both parts by 4), not 6:7 (adding 4 to each part).
• Not simplifying fully: 18:24 divided by 2 gives 9:12, but GCF is 6, so the simplest form is 3:4.
• Order matters: 3:5 and 5:3 are different ratios. Always check which quantity comes first.
• Mixing units: When setting up a proportion, both ratios must use the same units. Comparing 2 km to 5 miles and 4 km to 10 miles gives a wrong answer because the units do not match.
• Forgetting three-part ratios: A ratio like 1:2:3 has three parts that sum to 6, not 3. Each part is 1/6, 2/6 (1/3), and 3/6 (1/2) of the total.

Ratios in School Exams

Ratio problems appear in GCSE Maths (UK), SAT (US), and most national curricula from around age 11 onwards. Common question types include:

• Sharing in a given ratio: "Share 120 in the ratio 3:5" - total parts = 8, each part = 15, so the shares are 45 and 75.
• Finding a quantity from a ratio: "A:B = 4:7 and B = 35. Find A." - one part = 35/7 = 5, so A = 4 x 5 = 20.
• Combining ratios: If A:B = 2:3 and B:C = 3:5, then A:B:C = 2:3:5 (because B is 3 in both). If B values differ, scale to match - e.g. A:B = 2:3 and B:C = 4:5 becomes A:B:C = 8:12:15 (multiply first ratio by 4, second by 3).
• Proportion word problems: "If 5 workers paint a fence in 8 hours, how long would 4 workers take?" - this is inverse proportion: 5 x 8 = 40 worker-hours, so 40/4 = 10 hours.

The key distinction students need to grasp is between direct proportion (doubling one quantity doubles the other) and inverse proportion (doubling one quantity halves the other). Speed-distance-time problems often test inverse proportion, while recipe and mixing problems test direct proportion.

For more advanced fraction work beyond ratios, the fraction calculator handles addition, subtraction, multiplication, and division of fractions directly. The golden ratio calculator explores the 1:1.618 ratio and its applications in design.

Sources

• Wikipedia - Cross-multiplication
• Wikipedia - 16:9 aspect ratio
• Scientific American - A Brief History of Aspect Ratios
• Wells Fargo - Understanding Debt-to-Income Ratio
• Britannica - Golden Ratio