R3, R4 - Comparing quantities as fractions and ratios

R3, R4 - Comparing quantities as fractions and ratios

A fraction and a ratio can compare the same quantities in different forms. R3 shows how to express one quantity as a fraction of another, including answers greater than one. R4 adds ratio notation, order and simplest form, so you can make fair comparisons even when the units or numbers look different.

One quantity as a fraction of another

One ribbon is 45 cm45\,\mathrm{cm} long. Another is 0.6 m0.6\,\mathrm{m} long. What fraction of the second ribbon's length is the first ribbon's length? And what is their length ratio? The units are hiding a simple comparison.

You already know

Equivalent fractions have the same value. Divide numerator and denominator by a common factor to simplify, and make units agree before comparing measurements.

“Express 1212 as a fraction of 2020” means 12÷20=1220=3512\div20=\frac{12}{20}=\frac35. The quantity you are describing goes on top. The quantity you are comparing it with goes underneath. The second quantity is your reference amount; it need not be a total.

One quantity as a fraction of another

For quantities in the same units, with the reference quantity non-zero,

fraction=quantity being describedreference quantity\text{fraction}=\frac{\text{quantity being described}}{\text{reference quantity}}

The fraction is less than 11 if the first quantity is smaller, equal to 11 if they match, and greater than 11 if the first is larger.

Not on the formulae sheet: learn it.

The wording decides the order. First compare a smaller amount with a larger one.

Worked example 1: a fraction less than one

Question. Express 18 g18\,\mathrm{g} as a fraction of 30 g30\,\mathrm{g}. Give your answer in its simplest form. Non-calculator.

Plan. Put 1818 over 3030, then divide both by their highest common factor.

fraction=1830first quantity over reference=18÷630÷6divide both by 6=35simplify\begin{aligned} \text{fraction}&=\frac{18}{30} &&\text{first quantity over reference}\\ &=\frac{18\div6}{30\div6} &&\text{divide both by }6\\ &=\frac35 &&\text{simplify} \end{aligned}

Why is 3030 underneath? The question asks what fraction of 30 g30\,\mathrm{g} the 18 g18\,\mathrm{g} represents.

Answer. 35\frac35.

Check. 35×30=18\frac35\times30=18, and a smaller mass should give a fraction below 11.

Reverse only the order of the quantities. The new answer must exceed 11.

Worked example 2: a fraction greater than one

Question. Express 30 g30\,\mathrm{g} as a fraction of 18 g18\,\mathrm{g}. Give your answer in its simplest form. Non-calculator.

Plan. Put the quantity now being described on top, then simplify.

fraction=3018reference is now 18=30÷618÷6same common factor=53simplify\begin{aligned} \text{fraction}&=\frac{30}{18} &&\text{reference is now }18\\ &=\frac{30\div6}{18\div6} &&\text{same common factor}\\ &=\frac53 &&\text{simplify} \end{aligned}

Why keep this improper fraction? “A fraction of” does not mean the answer must be less than a whole.

Answer. 53\frac53, or 1231\frac23.

Check. 53×18=30\frac53\times18=30. The answer is greater than 11, as expected.

There is no need to turn a fraction into a rounded decimal.

Watch out

Wrong: put the smaller number on top every time. This would give 35\frac35 for both questions. Right: follow the wording; reversing the comparison changes 35\frac35 to 53\frac53.

Read “of” carefully in this new comparison.

Ratio order and simplest form

A ratio keeps the compared quantities in a stated order.

If there are 33 blue counters for every 55 orange counters, write blue:orange=3:5\text{blue}:\text{orange}=3:5. The colon separates the entries. It does not mean subtract. Reversing the labels gives orange:blue=5:3\text{orange}:\text{blue}=5:3.

[DIAGRAM: asset_name: The order in a ratio; asset_slug: aqa8300_r01_ratio_order; caption: Three blue counters and five orange counters give blue to orange as 3 to 5; description: Two labelled rows of equal-sized counters. The upper row contains three pale-blue counters and is labelled Blue: 3. The lower row contains five pale-clay counters and is labelled Orange: 5. A right-hand label states Blue : orange = 3 : 5 and beneath it Orange : blue = 5 : 3.]
Three blue counters and five orange counters give blue to orange as 3 to 5

Equivalent ratios

Ratios that describe the same relative amounts. For example, 3:53:5, 6:106:10 and 18:3018:30 are equivalent because every entry is multiplied by the same positive factor. A ratio is in simplest form when its entries are whole numbers with no common factor greater than 11.

Here is a useful surprise: 0.3:0.50.3:0.5 and 300:500300:500 make exactly the same comparison. A ratio records the relative size, not the size of the batch. Adding the same amount is different: 3:53:5 and 4:64:6 are not equivalent, since 35≠46\frac35\ne\frac46.

Continue this lesson.

You’re previewing NovaLearn without an account. Create a free account to read the rest.

Create free account