R6–R8, N11 - Ratios, fractions and proportion

R6–R8, N11 - Ratios, fractions and proportion

A ratio, a fraction and a straight-line equation can all describe the same relationship. R6 connects ratios with multipliers, R7 treats proportion as equality of ratios, and R8 links ratios to fractions and linear functions. N11 lets you use fractions inside ratio problems, including a fraction of just one group.

Multipliers as ratios and fractions

For every 22 litres of blue paint, a recipe needs 33 litres of white paint. How can the same recipe be written as a ratio, a fraction and an equation? We will finish with all three descriptions side by side.

You already know

A ratio keeps quantities in label order. Equivalent ratios scale every entry equally. In a straight-line equation y=mx+cy=mx+c, mm is the gradient and cc is the height when x=0x=0.

Start with a simple multiplier. If A's length is three times B's length, then whenever B has one length unit, A has three. Thus A:B=3:1A:B=3:1, and AB=3\frac AB=3. “Three times” is a multiplicative comparison; “three more” is an additive one.

Ratio and multiplier

For positive quantities AA and BB,

A:B=a:b⟺AB=ab⟺A=abBA:B=a:b \quad\Longleftrightarrow\quad \frac AB=\frac ab \quad\Longleftrightarrow\quad A=\frac ab B

The ratio and fraction must use the same order. The denominators must be non-zero.

Not on the formulae sheet: learn it.

A ratio need not compare whole-number multiples. A fractional multiplier can be turned into whole-number ratio entries.

Worked example 1: a whole-number multiplier

Question. Ribbon A is 33 times as long as ribbon B. Write A : B in its simplest form, and write A's length as a fraction of B's length. Non-calculator.

Plan. Compare A with one unit of B.

B=1 length unitchoose a referenceA=3 length unitsapply the multiplierA:B=3:1keep label orderAB=31same ordered comparison\begin{aligned} B&=1\text{ length unit} &&\text{choose a reference}\\ A&=3\text{ length units} &&\text{apply the multiplier}\\ A:B&=3:1 &&\text{keep label order}\\ \frac AB&=\frac31 &&\text{same ordered comparison} \end{aligned}

Why choose B as the reference? The statement tells us how many times B gives A.

Answer. A : B =3:1=3:1, and A is 31\frac31 of B's length.

Check. If B is 4 cm4\,\mathrm{cm}, A is 12 cm12\,\mathrm{cm}; 12:4=3:112:4=3:1.

Change only the multiplier from a whole number to a decimal. Clear the decimal to give a whole-number ratio.

Worked example 2: a non-integer multiplier

Question. Ribbon A is 2.52.5 times as long as ribbon B. Write A : B in its simplest form, and write A's length as a fraction of B's length. Non-calculator.

Plan. Write 2.5:12.5:1, then multiply both entries by 22.

A:B=2.5:1use one unit of B=5:2multiply both by 2AB=52same ordered fraction\begin{aligned} A:B&=2.5:1 &&\text{use one unit of B}\\ &=5:2 &&\text{multiply both by }2\\ \frac AB&=\frac52 &&\text{same ordered fraction} \end{aligned}

Why multiply both entries? This clears the decimal without changing the comparison.

Answer. A : B =5:2=5:2, and A is 52\frac52 of B's length.

Check. If B is 4 cm4\,\mathrm{cm}, A is 10 cm10\,\mathrm{cm}; 10:4=5:210:4=5:2.

A fractional comparison above one is perfectly valid: the first quantity is simply larger.

Watch out

Wrong: “A is 2.52.5 times B, so A : B =1:2.5=1:2.5.” Right: the first label is A, so write 2.5:1=5:22.5:1=5:2.

Keep the label order visible in this new multiplier.

Fractions of groups and of the whole

A fraction comparing two groups is different from the fraction of the whole in one group.

For red : blue =3:5=3:5, the red count is 35\frac35 of the blue count, but red counters make up 33+5=38\frac3{3+5}=\frac38 of all the counters. The denominator changes because the reference changes.

Conversely, if 27\frac27 of all counters are red and the rest are blue, imagine seven equal parts. Two are red and 7−2=57-2=5 are blue, so red : blue =2:5=2:5.

A group's fraction of a two-part whole

For a part : part ratio a:ba:b, with positive entries,

first fraction of whole=aa+b,second fraction of whole=ba+b\text{first fraction of whole}=\frac{a}{a+b}, \qquad \text{second fraction of whole}=\frac{b}{a+b}

If only a fraction ff of the first group is selected, the selected fraction of the whole is f×aa+bf\times\frac{a}{a+b}.

Not on the formulae sheet: learn it.

First identify a whole group, then take a fraction within that group.

Worked example 3: one group's fraction of the total

Question. A bag contains red and blue counters in the ratio 3:53:5. What fraction of all the counters are red? Non-calculator.

Plan. Compare the red parts with all eight parts.

3+5=8parts in the wholered fraction=38red parts over all parts\begin{aligned} 3+5&=8 &&\text{parts in the whole}\\ \text{red fraction}&=\frac38 &&\text{red parts over all parts} \end{aligned}

Why add the entries? The whole includes both colours.

Answer. 38\frac38.

Check. The blue fraction is 58\frac58, and 38+58=1\frac38+\frac58=1.

Add one condition: half the red counters carry a star. A fraction of one group is now needed.

Worked example 4: a fraction within one group

Question. Red : blue =3:5=3:5. Half of the red counters have a star; none of the blue counters do. What fraction of all counters have a star? Non-calculator.

Plan. Multiply the red fraction of the whole by the starred fraction of red.

red fraction=38three of eight partsstarred fraction=12×38half of the red group=316multiply fractions\begin{aligned} \text{red fraction}&=\frac38 &&\text{three of eight parts}\\ \text{starred fraction}&=\frac12\times\frac38 &&\text{half of the red group}\\ &=\frac3{16} &&\text{multiply fractions} \end{aligned}

Why not answer 12\frac12? Only half the red group is starred, not half the whole bag.

Answer. 316\frac3{16}.

Check. With 1616 counters there are 66 red and 1010 blue. Half the 66 red counters is 33, giving 33 starred out of 1616.

Choosing a convenient total is a useful check. Use a multiple that makes every count a whole number.

Watch out

Wrong: red : blue =3:5=3:5, so 35\frac35 of all counters are red. Right: 35\frac35 compares red with blue; 38\frac38 compares red with all counters.

The new question changes the fraction selected within the first group.

Proportion as equality of ratios

Two ordered pairs are in the same proportion when they have equal ratios.

Continue this lesson.

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