What Are Ratios and Proportions?

Think of a ratio as a way to compare two things using division. When you see something written as 4:5 or 4/5, that's a ratio showing how one quantity relates to another. It's like saying "for every 4 of this, there are 5 of that."

A proportion is when two ratios are equal to each other. For example, 4/5 = 20/25 is a proportion because both fractions have the same value when simplified. You can check this by cross-multiplying: 4 × 25 = 5 × 20, which gives you 100 = 100.

You'll see ratios everywhere in real life - from cooking (2 cups flour to 1 cup sugar) to speed (distance over time). The key thing to remember is that you can only compare quantities that have the same units.

Quick Check: Ratios can be written three ways: "a to b", "a:b", or "a/b" - they all mean the same thing!

Types of Proportion

Proportion comes in three main types that you'll encounter in maths problems. Understanding these will make solving equations much easier.

Direct proportion means when one thing increases, the other increases too. Think about buying sweets - the more you buy, the more you pay. If 2 sweets cost £1, then 4 sweets cost £2. The relationship stays consistent.

Inverse proportion works the opposite way - when one increases, the other decreases. Imagine sharing pizza slices: the more people at the party, the fewer slices each person gets. More people = fewer slices per person.

Continued proportion links three or more ratios together. If you have ratios a:b and c:d, you can create a continued proportion by making the middle terms match up.

Remember: Direct proportion uses multiplication $more = more$, while inverse proportion involves division $more = less$.

Essential Formulas and Terms

The ratio formula is straightforward: a:b becomes a/b, where 'a' is the antecedent (first term) and 'b' is the consequent (second term). So in 4:9, the number 4 is the antecedent and 9 is the consequent.

For proportions, when you have a:b::c:d, the terms 'a' and 'd' are called extremes, while 'b' and 'c' are the means. This gives you the equation a/b = c/d, which you can solve by cross-multiplying to get ad = bc.

There are several handy proportion properties with fancy Latin names, but the most useful ones are: if a:b = c:d, then you can flip both ratios $b:a = d:c$ or swap the means $a:c = b:d$. These tricks help solve complex problems.

The mean proportional between two numbers is found using the square root. For numbers a and b, the mean proportional is √(ab).

Top Tip: When solving proportions, cross-multiplication is your best friend - it turns ratios into simple equations!

Spotting the Difference

Knowing when to use ratios versus proportions will save you time in exams and homework. Ratios compare two quantities directly, while proportions show that two ratios are equal.

Look for keywords in word problems. "To every" usually signals a ratio problem - like "for every 3 boys, there are 5 girls." Meanwhile, "out of" typically means proportion - such as "out of 100 students, 60 are girls."

Ratios use a single colon (:) or slash (/) between numbers. Proportions use double colons (::) or equals signs (=) to show two ratios are the same. Remember: ratios are expressions, proportions are equations.

Special proportionals include the fourth proportional $finding the missing term in a:b = c:?$, third proportional, and mean proportional. These pop up regularly in exam questions.

Exam Hack: If you see "find the fourth proportional," set up the equation a:b = c:x and solve for x!

Useful Tricks and Rules

These ratio and proportion tricks will speed up your problem-solving and help you tackle trickier questions with confidence.

If u/v = x/y, then several useful relationships follow. You can cross-multiply to get uy = vx, flip both fractions $v/u = y/x$, or create new ratios like $u+v$/v = $x+y$/y. These transformations are incredibly handy for solving complex problems.

The componendo-dividendo rule is particularly useful: if u/v = x/y, then $u+v$/$u-v$ = $x+y$/$x-y$. This might look complicated, but it's just adding and subtracting the numerators and denominators in a specific way.

Duplicate ratios create patterns you should recognise. If a:b is your starting ratio, then a²:b² is the duplicate ratio, √a:√b is the sub-duplicate ratio, and a³:b³ is the triplicate ratio.

Memory Aid: When ratios get complex, always convert everything to the same units first - it prevents silly mistakes!

Practice Examples Made Simple

Let's work through some real-world examples that show how ratios and proportions solve everyday problems.

Example 1: Are 4:5 and 8:10 in proportion? Convert both to decimals: 4/5 = 0.8 and 8/10 = 0.8. Since they're equal, yes they're in proportion! But 8:10 and 7:10 aren't in proportion because 0.8 ≠ 0.7.

Example 2: In an office of 45 people, 25 are female. The ratio of females to males is 25:20, which simplifies to 5:4. Always simplify ratios to their lowest terms by dividing by the highest common factor.

Example 3: If Rohan earns £12,000 monthly and Anish earns £191,520 yearly, both spending £9,960 monthly, what's their savings ratio? Rohan saves £2,040 monthly, Anish saves £6,000 monthly (after converting yearly to monthly). The ratio is 2040:6000 = 17:50.

Pro Tip: Always check your units match before comparing - convert yearly figures to monthly, metres to centimetres, etc.!

More Problem-Solving Examples

These examples show different types of ratio and proportion problems you'll encounter in your studies.

Finding costs using ratios: If 20 tons of iron costs £600,000, what does 560kg cost? First convert: 20 tons = 20,000kg. So 1kg costs £600,000 ÷ 20,000 = £30. Therefore 560kg costs £30 × 560 = £16,800.

Working with dimensions: A field is 50m long and 15m wide. The ratio of length to breadth is 50:15 = 10:3 (dividing both by 5). Always simplify ratios to make them easier to work with.

Unit conversion problems: To find the ratio of 90cm to 1.5m, convert to the same units first. 1.5m = 150cm, so the ratio becomes 90:150 = 3:5. This type of question tests whether you remember to convert units.

Finding unknown numbers: If two numbers are in the ratio 2:3 and their sum is 60, call them 2x and 3x. Then 2x + 3x = 60, so 5x = 60, giving x = 12. The numbers are 24 and 36.

Success Strategy: Set up equations using 2x, 3x, 4x etc. when you know ratios but not actual values - it makes algebra much simpler!