What is the Rule of Three?

The Rule of Three is a mathematical method to calculate an unknown value when three values of a ratio are known. It is one of the most important tools in everyday mathematics, used in many areas such as shopping, cooking, finance, and technology.

How do you calculate the Rule of Three?

For the proportional Rule of Three: 1) Write down the known values (a units = b value). 2) Divide by a to get the value for 1 unit. 3) Multiply by the desired quantity c. The formula is: x = (b × c) / a.

What is the difference between proportional and inverse?

In proportional Rule of Three, both quantities increase together (twice as much → twice the price). In inverse Rule of Three, one quantity increases while the other decreases (twice as many workers → half the time). The inverse formula is: x = (a × b) / c.

When do you need the inverse Rule of Three?

You need the inverse Rule of Three when one value increases and the other decreases as a result. Typical examples: more workers → less time, higher speed → shorter travel time, more machines → less production time per unit.

What is a compound Rule of Three?

A compound Rule of Three is used when more than two quantities are involved. It is solved step by step: first adjust the first quantity (proportional or inverse), then the second. The result from step 1 becomes the starting value for step 2.

Can you solve percentages with the Rule of Three?

Yes! Percentage calculation is a special case of the Rule of Three. Example: 100% = €500, find: 15%. Solution: €500 ÷ 100 × 15 = €75. The Rule of Three is often simpler because you don't need to distinguish between base value, percentage value, and rate.

In which grade do you learn the Rule of Three?

The Rule of Three is typically introduced in grade 6 or 7. The proportional version comes first, the inverse version usually follows one school year later. The compound Rule of Three is often covered in grade 8 or 9.

How do I tell if it's proportional or inverse?

Ask yourself: 'If one quantity increases, does the other increase or decrease?' If it increases → proportional. If it decreases → inverse. Example: buying more apples → higher price (proportional). More people share a pizza → less per person (inverse).

What is the difference between the Rule of Three and a ratio equation?

Mathematically, both are equivalent. The Rule of Three solves the problem step by step (calculate for 1, then scale up), the ratio equation sets a:b = c:x and solves for x. The Rule of Three is more intuitive, while the ratio equation is more concise.

How do you verify a Rule of Three calculation?

For proportional Rule of Three, check that the ratio stays the same: a/b must equal c/x. For inverse Rule of Three, check that the products are equal: a × b must equal c × x. If the check passes, your result is correct.

Rule of Three Exercises with Solutions: 20 Practice Problems | Rule of Three Calculator

Rule of Three Exercises with Solutions: 20 Practice Problems

Practice makes perfect! Here are 20 Rule of Three exercises with complete solution paths — from easy to challenging. Perfect for students and anyone wanting to refresh their skills.

Simple Proportional Problems

**Problem 1:** 4 notebooks cost $6.00. How much do 10 cost?

Solution: 1 notebook = $6.00 / 4 = $1.50 -> 10 notebooks = $1.50 x 10 = $15.00

**Problem 2:** 200 g of cheese costs $3.60. How much do 350 g cost?

Solution: 1 g = $3.60 / 200 = $0.018 -> 350 g = $0.018 x 350 = $6.30

**Problem 3:** A car uses 7 liters per 100 km. How much does it use for 280 km?

Solution: 1 km = 7 / 100 = 0.07 L -> 280 km = 0.07 x 280 = 19.6 liters

**Problem 4:** 15 m of fabric costs $67.50. How much do 23 m cost?

Solution: 1 m = $67.50 / 15 = $4.50 -> 23 m = $4.50 x 23 = $103.50

**Problem 5:** 8 bread rolls cost $2.80. How much do 20 cost?

Solution: 1 roll = $2.80 / 8 = $0.35 -> 20 rolls = $0.35 x 20 = $7.00

Medium Difficulty Proportional Problems

**Problem 6:** A recipe for 4 people needs 250 g of butter. How much for 7 people?

Solution: 1 person = 250 / 4 = 62.5 g -> 7 people = 62.5 x 7 = 437.5 g

**Problem 7:** 12 workers build a wall in 5 days. How long for 3 walls with 12 workers?

**Solution:** 1 wall = 5 days -> 3 walls = 5 x 3 = **15 days** (proportional: more walls = more time)

**Problem 8:** 3.5 kg of flour costs $2.45. How much do 12 kg cost?

Solution: 1 kg = $2.45 / 3.5 = $0.70 -> 12 kg = $0.70 x 12 = $8.40

Inverse Problems

**Problem 9:** 6 workers need 10 days. How long do 15 workers need?

Solution: Total effort = 6 x 10 = 60 -> 15 workers = 60 / 15 = 4 days

**Problem 10:** One pump empties a pool in 8 hours. How long do 4 pumps take?

Solution: Total effort = 1 x 8 = 8 -> 4 pumps = 8 / 4 = 2 hours

**Problem 11:** 5 machines produce a batch in 12 hours. How long do 20 machines take?

Solution: Total effort = 5 x 12 = 60 -> 20 machines = 60 / 20 = 3 hours

**Problem 12:** Supplies last 8 people for 15 days. How long for 12 people?

Solution: Total supply = 8 x 15 = 120 -> 12 people = 120 / 12 = 10 days

Challenging Problems

**Problem 13:** 250 ml of paint covers 4 m2. How much for 25 m2?

**Solution:** 1 m2 = 250 / 4 = 62.5 ml -> 25 m2 = 62.5 x 25 = **1,562.5 ml** (approximately 1.6 liters)

**Problem 14:** At 60 km/h a journey takes 45 minutes. How long at 90 km/h?

**Solution:** Total distance = 60 x 45 = 2,700 -> 90 km/h = 2,700 / 90 = **30 minutes** (inverse)

**Problem 15:** 100 g of chocolate has 540 kcal. How many kcal in 35 g?

Solution: 1 g = 540 / 100 = 5.4 kcal -> 35 g = 5.4 x 35 = 189 kcal

Compound Problems

**Problem 16:** 4 workers produce 120 parts in 6 hours. How many parts do 6 workers make in 8 hours?

Solution: Step 1 (workers, proportional): 120 x 6/4 = 180. Step 2 (hours, proportional): 180 x 8/6 = 240 parts

**Problem 17:** 3 machines need 10 hours for 500 pieces. How many do 5 machines make in 8 hours?

Solution: Step 1 (machines, proportional): 500 x 5/3 = 833. Step 2 (hours, proportional): 833 x 8/10 = 667 pieces

**Problem 18:** 8 lamps run 5 hours on 10 batteries. How many batteries for 12 lamps for 3 hours?

Solution: Step 1 (lamps, proportional): 10 x 12/8 = 15. Step 2 (hours, proportional): 15 x 3/5 = 9 batteries

**Problem 19:** 6 printers print 2,400 pages in 4 hours. How many pages do 9 printers print in 3 hours?

Solution: Step 1: 2,400 x 9/6 = 3,600. Step 2: 3,600 x 3/4 = 2,700 pages

**Problem 20:** 5 chefs cook for 120 guests in 3 hours. How many chefs for 200 guests in 2 hours?

Solution: Step 1 (guests, proportional): 5 x 200/120 = 8.33. Step 2 (hours, inverse): 8.33 x 3/2 = 12.5 -> 13 chefs

Tips for Practice

- Read each problem twice and identify: What is given? What is sought?

- Decide first: proportional or inverse?

- Always write out all three steps — even if you could do it mentally.

- Verify your answer — it builds confidence.

- Use our Rule of Three calculator to check your work!