Ratio

1. **Enter the ratio terms** – Type the values of A and B (and optionally C if solving a proportion) into the input fields.
2. **Choose your operation** – Select whether you want to Simplify, Scale to a total, or Solve for a missing value.
3. **Enter the target (if scaling)** – If scaling, provide the target total T that the ratio parts should add up to.
4. **Click Calculate** – The calculator instantly returns the simplified or scaled ratio, the GCD used, and the scale factor.
5. **Read the result** – Review the output ratio and the step-by-step breakdown to understand exactly how the result was reached.

Simplified: (A/GCD) : (B/GCD) | Scaled: (A×k) : (B×k), where k = T/(A+B) | Missing value: D = (B×C)/A

• A — The first term of the original ratio.
• B — The second term of the original ratio.
• GCD(A, B) — The Greatest Common Divisor of A and B — the largest integer that divides both A and B evenly.
• T — The target total (sum of all ratio parts) used when scaling a ratio to a specific quantity.
• k — The scale factor; calculated as T divided by the sum of the original ratio parts (A + B).
• C — The first term of an equivalent ratio when solving a proportion A : B = C : D.
• D — The unknown fourth term in a proportion, solved using cross-multiplication.

**Step 1 – Simplify:**
GCD(36, 48) = 12
Simplified = (36 ÷ 12) : (48 ÷ 12) = 3 : 4

**Step 2 – Scale to total T = 35:**
Sum of simplified parts = 3 + 4 = 7
Scale factor k = 35 ÷ 7 = 5
Scaled values = (3 × 5) : (4 × 5) = 15 : 20

Result: Simplified ratio: **3 : 4** | Scaled to total 35: **15 : 20** (which also simplifies to 3 : 4, confirming equivalence).

The simplified ratio 3 : 4 is the irreducible form of 36 : 48 — both terms share no common factor larger than 1 after dividing by the GCD of 12. When scaled to a total of 35, the parts become 15 and 20, which together sum to 35 and maintain the exact same proportional relationship. You can verify equivalence by simplifying 15 : 20 back to 3 : 4 (GCD = 5).

Understanding Ratios

A ratio is a way of comparing two or more quantities by division. The ratio A : B tells you how many times one value contains or is contained by the other. Ratios appear in everyday life — recipes, maps, finance, engineering, and more.

Simplifying Ratios

A ratio is in its simplest form (or lowest terms) when the GCD of all its terms equals 1. To simplify, divide every term by the GCD. For example, 18 : 24 simplifies to 3 : 4 because GCD(18, 24) = 6.

Equivalent Ratios

Two ratios are equivalent if one can be obtained from the other by multiplying or dividing all terms by the same non-zero number. 3 : 4, 6 : 8, and 15 : 20 are all equivalent.

Proportions

A proportion is a statement that two ratios are equal: A : B = C : D. This is often written as a fraction equation A/B = C/D and solved using cross-multiplication: A × D = B × C.

Part-to-Part vs. Part-to-Whole

• Part-to-part ratio (e.g., 3 : 4) compares one subset to another.
• Part-to-whole ratio (e.g., 3 : 7) compares one subset to the entire group.

Applications

• Recipes: Scaling ingredient quantities while keeping flavour balance.
• Maps & Scale Models: A map scale of 1 : 50,000 means 1 cm on the map equals 50,000 cm in real life.
• Finance: Debt-to-equity ratio, price-to-earnings ratio.
• Mixing & Dilution: Combining chemicals, paint colours, or concrete at the right proportions.

• **Forgetting to simplify fully** – Dividing by a common factor that isn't the GCD leaves the ratio in a partially reduced but not fully simplified form. Always find the GCD, not just any common factor.
• **Confusing part-to-part with part-to-whole** – A ratio of 3 : 4 means 3 parts to 4 parts (7 total), not 3 out of 4. Mixing these up leads to incorrect fraction conversions.
• **Scaling by adding instead of multiplying** – To double a ratio, multiply every term by 2. Adding a constant to each term changes the ratio's value entirely (e.g., 1 : 2 + 1 : 2 gives 2 : 3, not 2 : 4).
• **Using different units** – Both terms in a ratio must be in the same unit before comparing. Convert first (e.g., cm to mm) or the ratio will be meaningless.
• **Rounding intermediate steps** – When solving proportions, round only the final answer. Rounding during calculation introduces cumulative errors.

• Greatest Common Divisor – Khan Academy — Khan Academy
• Ratios and Proportions – Khan Academy — Khan Academy
• Common Core State Standards – Ratios and Proportional Relationships (Grade 6) — Common Core State Standards Initiative

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