Ratios and Rates

Ratios and rates are the same underlying idea: comparing two quantities. A ratio compares parts (3:2). A rate compares a quantity to a unit (60 km/hr). The techniques overlap heavily, so they belong in one lesson.

Part-to-Part vs Part-to-Whole

Every ratio question is one of these two types. Spot which one you need before calculating.

Example: a class has 12 boys and 8 girls.

Part-to-part: boys : girls = 12 : 8 = 3 : 2 (comparing one group to another)
Part-to-whole: boys / total = 12 / 20 = 3/5 = 60% (comparing one group to the total)

Simplifying Ratios

Convert to a decimal on the calculator, then recognise the fraction:

Problem: what is the ratio of 480 to 360?

1. 480 / 360 = 1.333…
2. Recognise 1.333 = 4/3
3. Ratio = 4 : 3

If you don't recognise the decimal, divide both numbers by the smaller one: 480 / 360 = 1.333 and 360 / 360 = 1 → ratio = 1.333 : 1 = 4 : 3.

The Delta Shortcut

When a ratio question gives you the difference between two groups, this avoids the algebra:

Problem: boys and girls are in ratio 9 : 1. There are 40 more boys than girls. How many boys?

Standard: let girls = x, boys = 9x. 9x − x = 40 → 8x = 40 → x = 5. Boys = 45.

Delta shortcut: ratio difference = 9 − 1 = 8 parts. 8 parts = 40 people → 1 part = 5 people. Boys = 9 parts = 45.

Same answer, no algebra.

Worked Example: Mixing Problem

Question: A paint mix uses red, blue, and white paint in the ratio 3:5:2. A decorator needs 4 litres of the mix. How many millilitres of blue paint does she need?

1. Total parts = 3 + 5 + 2 = 10
2. Blue = 5 / 10 = 1/2 of the mix
3. 1/2 of 4 litres = 2 litres = 2,000 ml

Calculator barely needed - mental maths handles this one. Time: ~12 sec.

If the ratio were less clean (say 3:7:2 = 12 parts), you'd do 4000 × 7 / 12 = 2,333 ml.

Worked Example: Ratio with Speed/Distance/Time

Question: Two cars start from the same point. Car A drives north at 80 km/h, Car B drives south at 60 km/h. After 2 hours, what's the ratio of the distance between them to the distance Car B has travelled?

1. Car A distance = 80 × 2 = 160 km
2. Car B distance = 60 × 2 = 120 km
3. Distance between them = 160 + 120 = 280 km (opposite directions)
4. Ratio = 280 : 120 → divide both by 40 → `7 : 3`

This combines speed-distance-time with ratio simplification. Time: ~20 sec.

Speed-distance-time is the most common rate question type in QR. If you learn one rate formula, learn this one.

The Speed Triangle

Picture D (distance) on top with S (speed) and T (time) below it side-by-side. Cover the variable you want - the remaining two show you the formula.

The Unit Mismatch Trap

The number one error in speed/distance/time: mismatched units. Always check before calculating.

Problem: a car travels 150 km at 60 km/h. How many minutes does the journey take?

Wrong: T = D/S = 150/60 = 2.5. Answer: 2.5 (wrong unit).
Right: T = D/S = 150/60 = 2.5 hours = 2 hours 30 minutes = 150 min.

The question asked for minutes, not hours.

Key Time Conversions

Memorise these to skip calculator use on time conversions:

Worked Example: Speed/Distance/Time

Question: A cyclist travels 24 km in 1 hour 20 minutes. What is the cyclist's average speed in km/h?

1. Convert time to hours: 1 hr 20 min = 1 + 20/60 = 1 + 1/3 = 1.333 hr
2. Speed = Distance / Time = 24 / 1.333 = 18 km/h

Calculator: 24 / 1.333 =. Time: ~15 sec.

Speed is just one type of rate. Any quantity measured "per unit" is a rate, and the logic is identical. "Per" = division:

• Cost per item = Total cost / Number of items
• Litres per hour = Total litres / Number of hours
• People per km² = Population / Area
• Calories per gram = Total calories / Grams

Rate Problems: The Setup

This is the speed triangle generalised - Total on top, Rate × Units below:

Benchmark Fractions for Rate Problems

When dividing produces an ugly decimal, these help you recognise the answer:

Trap 1: Part-to-part vs part-to-whole confusion

Ratio 3 : 2 does not mean 3/5 and 2/5 unless the question asks for proportions. If it asks "ratio of A to B", give 3 : 2.

Trap 2: Mismatched units in speed questions

Speed in km/h but distance in metres? Convert first. Speed in m/s but time in minutes? Convert first.

Trap 3: "Times as many" vs "more than"

• "3 times as many" = multiply by 3
• "3 more than" = add 3

Different operations, wildly different numbers.

Trap 4: Average speed over a journey

Average speed = total distance / total time - not the average of two speeds.

60 km/h for 60 km + 30 km/h for 60 km: Time 1 = 60/60 = 1 hr. Time 2 = 60/30 = 2 hr.
Total = 120 km / 3 hr = 40 km/h, not (60 + 30)/2 = 45 km/h.

Next lesson: 3.4 Averages & Statistics - mean, median, range, and the delta shortcut that eliminates most of the arithmetic.