Ratios, percentages, and rates

Why this matters

Ratios, percentages and rates are the most common calculations in working life. A recipe for a cleaning solution is a ratioA comparison of two or more quantities, such as 1 : 2 : 3 parts of cement, sand and gravel.. A price increase, a battery's charge level and a test's pass rateA ratio of two quantities with different units, usually written "per" something, such as litres per minute. are percentages. Speed, flow, data transfer and dosage are rates. You do not need advanced maths for any of them, but you do need to set them up correctly, because small set-up mistakes give answers that look reasonable and are wrong.

Ratios: comparing parts

A ratio compares two quantities. A concrete mix of 1 part cement to 2 parts sand to 3 parts gravel is written 1 : 2 : 3. The total is 6 parts, so cement is 1/6 of the mix.

Example: you need 30 kg of that mix.

• One part = 30 kg / 6 = 5 kg
• Cement = 1 × 5 = 5 kg, sand = 2 × 5 = 10 kg, gravel = 3 × 5 = 15 kg

Ratios scale: double every part and the ratio stays the same. That is how you resize a recipe, a solution or a drawing. A map scale of 1 : 50,000 means 1 cm on the map is 50,000 cm (500 m) on the ground.

Percentages: parts per hundred

A percentage is a ratio out of 100. 25% means 25 per 100, which is 0.25 as a decimal. To find a percentage of something, convert to a decimal and multiply: 15% of 80 = 0.15 × 80 = 12.

Percent changeThe difference between a new and an old value, divided by the old value and multiplied by 100. compares a new value with an old one:

percent change = (new - old) / old × 100

Example: a part's price goes from $50 to $60.

(60 - 50) / 50 × 100 = 10 / 50 × 100 = 20% increase

If it then falls back from $60 to $50:

(50 - 60) / 60 × 100 = -10 / 60 × 100 ≈ -16.7%

A 20% rise followed by a 16.7% fall gets you back to the start. Percent changes are always measured from the old value, so up and down are not symmetric. This trips people up all the time: a 50% loss needs a 100% gain to recover.

Percentage points are not percent

If a pass rate goes from 40% to 50%, it rose by 10 percentage points, but by 25 percent (10 / 40 = 0.25). Both statements are true, and they sound very different. In reports, say which one you mean.

Rates: one quantity per another

A rate is a ratio with different units, usually "per" something: kilometres per hour, litres per minute, megabits per second, milligrams per kilogram. A unit rateThe amount for exactly one unit of the second quantity, such as 30 L/min or $9/kg. is the amount for one unit of the second quantity.

Example: a pump moves 450 litres in 15 minutes.

450 L / 15 min = 30 L/min

Now you can answer other questions. How long for 1,200 litres? 1,200 / 30 = 40 minutes.

Example: a download of 600 megabytes takes 2 minutes. Megabytes per second = 600 / 120 = 5 MB/s. Internet speeds are usually given in megabits per second, and 1 byte = 8 bits, so that is 40 Mbit/s. Units matter here too.

Rates let you compare options fairly. A 2 kg bag of fasteners for $18 costs $9/kg; a 500 g box for $5 costs $10/kg. The bigger bag is cheaper per kilogram, even though its price tag is higher.

Common mistakes

• Dividing by the wrong value in percent change. Always divide by the old (starting) value.
• Adding percentages in a row. A 10% rise then another 10% rise is a 21% total rise (1.1 × 1.1 = 1.21), not 20%.
• Confusing percent and percentage points. Say which one you mean.
• Mixing units inside a rate. Convert 15 minutes to 0.25 hours before you calculate kilometres per hour.
• Forgetting the total in a ratio. In 1 : 2 : 3, cement is 1/6 of the mix, not 1/3.

Quick recap

• A ratio compares parts; divide by the total number of parts to share a quantity.
• Percent change = (new - old) / old × 100, measured from the old value.
• Percentage points and percent are different things.
• A rate is "per" something; a unit rate makes comparisons easy.

Try it

Open the Do tab. A small JavaScript calculator already has a working unit-rate function. You will write the one line that calculates percent change, then the checks will test it with price rises and falls. No programming experience is needed.