Ratio, Proportion and Rates

What a ratio is, and how to simplify one

A ratio compares quantities of the same kind. Writing $20:30:40$ says the three amounts are in the same proportion as $2:3:4$. Because a ratio compares like with like, the numbers carry no units: $20:30:40$ is the same ratio whether the amounts are grams, minutes or dollars.

To simplify a ratio, divide every part by the same number until the parts share no common factor other than 1. $20:30:40$ has a common factor of 10, so dividing every part by 10 gives $2:3:4$. The numbers have been scaled down, but the relationship between them has not changed.

The units must match before you simplify. A ratio of 250 g to 1.5 kg is not 250 to 1.5: convert first so both parts are in grams, giving $250:1500$, which simplifies to $1:6$. Cambridge expects whole numbers in the simplest form wherever possible, so multiply out decimals and fractions first: $0.5:2$ becomes $1:4$, and $1.5:2.5$ becomes $3:5$.

Dividing by the highest common factor gets you there in one step. For $45:60:75$ the highest common factor is 15, and dividing every part by 15 gives $3:4:5$. If you cannot spot the highest common factor, divide by anything that goes into all the parts and repeat until nothing else does.

Dividing a quantity in a given ratio

Dividing in a ratio means splitting a total into parts whose sizes are in that ratio. The method has three steps: add the numbers in the ratio to find the number of equal parts, divide the total by that number to find the value of one part, then multiply by each number in the ratio.

Share 90 sweets in the ratio $2:3:4$. The number of parts is $2+3+4=9$, so one part is $90÷9=10$ sweets. The shares are $2\times 10=20$, $3\times 10=30$ and $4\times 10=40$. The check is always the same: the shares must add back to the amount you started with, and $20+30+40=90$.

For a two-part ratio, share 3.5 kg in the ratio $4:3$. There are 7 parts, so one part is $3.5÷7=0.5$ kg and the shares are 2 kg and 1.5 kg. Notice that the answer keeps the unit of the original quantity - the ratio itself has none, so the shares of a mass are masses.

The same method works when the total is money. Share $\pounds 60$ in the ratio $2:3$: there are 5 parts, one part is $\pounds 12$, so the shares are $\pounds 24$ and $\pounds 36$. Keep the ratio in the order the question gives it, because a question that asks for the larger share is marked on the share you name, not just on the arithmetic.

• 90 sweets in $2:3:4$: 9 parts, so 1 part is 10 - shares 20, 30 and 40
• $\pounds 60$ in $2:3$: 5 parts, so 1 part is $\pounds 12$ - shares $\pounds 24$ and $\pounds 36$
• 3.5 kg in $4:3$: 7 parts, so 1 part is 0.5 kg - shares 2 kg and 1.5 kg
• 2.4 m in $1:5$: 6 parts, so 1 part is 0.4 m - shares 0.4 m and 2.0 m
• Always check that the shares add back to the quantity you started with

Ratio in context: recipes, map scales and best value

Cambridge names three contexts for proportional reasoning: adapting recipes, using map scales and deciding which of two offers is better value. All three are the same skill, which is keeping a ratio fixed while the total changes.

In a recipe, every ingredient is scaled by the same factor. A recipe for 12 flapjacks uses 300 g of oats; for 20 flapjacks the factor is $20÷12=\frac{5}{3}$, so the oats needed are $300\times \frac{5}{3}=500$ g. Scaling only the ingredient named in the question is a common error, because the other ingredients may be needed for a later part of the same question.

A map scale such as $1:50000$ is a ratio: 1 cm on the map stands for 50 000 cm on the ground. Convert to a sensible unit first - 50 000 cm is 500 m - so 4 cm on the map is 2 km on the ground and 7.5 cm is 3.75 km. Divide the map measurement by the scale when you are finding a real distance, and multiply when you are finding how long a real distance would be on the map.

Best value questions compare cost per unit. If 3 kg of potatoes costs 4.50 dollars and 5 kg costs 7.20 dollars, the unit prices are $4.50÷3=1.50$ dollars per kg and $7.20÷5=1.44$ dollars per kg, so the 5 kg bag is better value. Comparing the two total prices tells you nothing, because the bags are different sizes.

Proportion and the unitary method

Two quantities are in direct proportion when multiplying one by a number multiplies the other by the same number. If 5 identical pens cost 3.75 dollars, then 10 pens cost 7.50 dollars and 20 pens cost 15.00 dollars, because the cost per pen has not changed.

The unitary method finds the value of one unit first and then multiplies. From 5 pens at 3.75 dollars, one pen costs $3.75÷5=0.75$ dollars, so 8 pens cost $8\times 0.75=6.00$ dollars. This is the safest method when the question does not scale by a whole number, such as asking for the cost of 7 pens.

The same method handles currency, pay and recipe questions, which is why ratio and proportion are one topic rather than three. Find the value of one hour, one litre or one serving, then multiply by the number you actually need. Keep the unrounded value on the calculator display and round only at the end.

Not every proportion is direct. If 4 workers take 9 days to build a wall, 12 workers do not take 27 days - more workers means less time. The total work is $4\times 9=36$ worker-days, so 12 workers take $36÷12=3$ days. When one quantity goes up while the other goes down, the product stays constant instead of the ratio.

Rates: pay, currency, flow, fuel, density and pressure

A rate compares two different quantities, which is what separates it from a ratio. 14.50 dollars per hour, 12 km per litre and 200 people per square kilometre are all rates, and the word per tells you that a division has happened. Someone paid 14.50 dollars an hour who works 7 hours 30 minutes earns $14.50\times 7.5=108.75$ dollars.

Currency conversion is a rate. The question always gives you the rate, so use the one in the question and never one you remember from the news. At 1 pound = 1.25 dollars, 240 pounds is $240\times 1.25=300$ dollars, and 450 dollars is $450÷1.25=360$ pounds. Decide the direction by asking whether the answer should be a bigger or a smaller number than the one you started with.

Flow rate and fuel consumption use the same idea. A tap that fills 12 litres in 4 minutes has a rate of $12÷4=3$ litres per minute, so filling 45 litres takes 15 minutes. A car using 45 litres over 540 km does $540÷45=12$ km per litre, and at 1.40 dollars per litre that journey costs $45\times 1.40=63.00$ dollars.

Density is mass divided by volume, and pressure is force divided by area. A metal block of mass 240 g and volume 30 cm${}^3$ has a density of $240÷30=8$ g/cm${}^3$, and a force of 600 N over an area of 0.5 m${}^2$ gives a pressure of $600÷0.5=1200$ N/m${}^2$. Population density is people divided by area: 96 000 people spread over 480 km${}^2$ is $96000÷480=200$ people per km${}^2$.

Average speed: distance, time and units

The syllabus requires you to know the speed, distance and time relationship, and it is not given in the question. Average speed is total distance divided by total time, $s=\frac{d}{t}$. Rearranged, $d=s\times t$ and $t=\frac{d}{s}$. If you forget which is which, remember that speed is how far you travel in one unit of time.

Units decide the answer. A distance in km with a time in hours gives a speed in km/h, and metres with seconds gives m/s. A time of 2 hours 20 minutes is $2\frac{1}{3}$ hours, so 84 km in that time is $84÷2\frac{1}{3}=36$ km/h. Convert minutes to a fraction of an hour before dividing: 15 minutes is 0.25 hours, 30 minutes is 0.5 hours and 45 minutes is 0.75 hours.

Average speed is not the average of the speeds. A driver covers 180 km at 90 km/h, which takes 2 hours, and then 45 km at 45 km/h, which takes 1 hour. The whole journey is 225 km in 3 hours, so the average speed is $225÷3=75$ km/h - while the mean of 90 and 45 is only 67.5 km/h. The mean is wrong because the two speeds were not held for equal times.

Converting between speed units is worth learning as a rule. To turn km/h into m/s, multiply by $1000÷3600$, which is the same as dividing by 3.6, so 72 km/h is 20 m/s. To go the other way, multiply by 3.6: 15 m/s is 54 km/h.

Part to part, part to whole: the two ways a ratio is used

A ratio such as $2:5$ describes the parts, not the whole. The whole is $2+5=7$ parts, so the first share is $\frac{2}{7}$ of the total and the second is $\frac{5}{7}$. Writing $\frac{2}{5}$ because the ratio says $2:5$ is the mistake that costs the most marks in this topic.

Different questions give you different pieces of the picture. If you are given the total, divide it by the number of parts. If you are given one share, divide that share by its own number of parts. If you are given the difference between the shares, divide by the difference between the ratio numbers.

In the ratio $3:5$ the shares differ by $5-3=2$ parts. If a question says the two amounts differ by 8, then 2 parts is 8, so 1 part is 4 and the shares are 12 and 20. The same ratio with a total of 48 gives $48÷8=6$ for each part and shares of 18 and 30.

The same reasoning answers questions where an amount is missing. If A : B = $3:4$ and A is 21, then 3 parts is 21, so 1 part is 7 and B is 28. Name the quantity that one part represents before you divide, because that one sentence is what keeps the arithmetic honest.

• Total given: divide the total by the sum of the ratio numbers
• One share given: divide that share by its own number of parts
• Difference given: divide the difference by the difference between the ratio numbers
• Check the answer is in the given ratio and that the shares add to the total