R5 - Ratio problems and best buys

R5 - Ratio problems and best buys

Ratios help with recipes, mixtures, conversions and shopping decisions. This part of R5 shows how to choose the quantity that corresponds to one part, scale it, and compare like with like. You will also use a known share or difference to recover missing amounts.

Using a conversion ratio

A 500 g500\,\mathrm{g} bag costs £1.80 and a 750 g750\,\mathrm{g} bag costs £2.55. Is the larger bag better value? The prices alone cannot settle it. By the end you will compare the same amount of food in each offer.

You already know

Equivalent ratios scale both entries together. To share a total in a ratio, add the part counts first. Make units agree before comparing measurements.

A conversion rate gives a pair of matching quantities. For a currency question, use the supplied rate and any stated fees. Do not replace it with a rate you happen to know.

Here the given rate is £1 =\1.25,withnofees.Thedollaramountforanypoundamountis, with no fees. The dollar amount for any pound amount is 1.25$ times as much. This particular rate is given in the question; it is not a formula-sheet fact.

Worked example 1: pounds to dollars

Question. A currency service uses £1 =\1.25$, with no fees. Work out how many dollars £48 buys. Non-calculator.

Plan. Scale the supplied one-pound amount by 4848.

48×1.25=48+484one and a quarter per pound=48+12=60total dollars\begin{aligned} 48\times1.25&=48+\frac{48}{4} &&\text{one and a quarter per pound}\\ &=48+12=60 &&\text{total dollars} \end{aligned}

Why multiply? Every pound buys another \1.25$.

Answer. \60$.

Check. 60÷48=1.2560\div48=1.25 dollars per pound.

Reverse only the direction. The question now starts with dollars, so divide by the dollars corresponding to one pound.

Worked example 2: dollars to pounds

Question. At £1 =\1.25,withnofees,howmanypoundsdoes, with no fees, how many pounds does $60$ buy? Non-calculator.

Plan. Count how many groups of \1.25fitintofit into$60$.

60÷1.25=6000÷125use cents in both amounts=48number of one-pound groups\begin{aligned} 60\div1.25&=6000\div125 &&\text{use cents in both amounts}\\ &=48 &&\text{number of one-pound groups} \end{aligned}

Why divide rather than multiply? This reverses the pounds-to-dollars conversion.

Answer. £48.

Check. 48×1.25=6048\times1.25=60, returning to the dollar amount.

A comparison involving two currencies must use a common currency first.

Watch out

Wrong: \60isworthmorethan£50becauseis worth more than £50 because60>50.Atthestatedrate,. At the stated rate, $60$ is £48, so it is worth less. Compare the amounts after converting, not their bare numbers.

Use the same rate in the new reverse conversion.

Scaling a recipe

A recipe keeps its ingredient ratios only when every ingredient is multiplied by the same factor.

If a recipe serves 44 people and you need 1010 servings, the multiplier is 10÷4=2.510\div4=2.5. It need not be a whole number. Assume that the ingredients scale with the servings; do not scale oven temperature or cooking time by that factor.

Scaling quantities

When quantities keep the same ratio,

scale factor=new matching amountoriginal matching amount\text{scale factor}=\frac{\text{new matching amount}}{\text{original matching amount}}

Multiply every other original quantity by this same factor.

Not on the formulae sheet: learn it.

The matching amounts here are the numbers of servings.

Worked example 3: one ingredient

Question. A recipe for 44 people uses 300 g300\,\mathrm{g} of flour. Work out the flour needed for 1010 people. Non-calculator.

Plan. Find the servings multiplier, then apply it to the flour.

10÷4=2.5servings multiplier300×2.5=750grams of flour\begin{aligned} 10\div4&=2.5 &&\text{servings multiplier}\\ 300\times2.5&=750 &&\text{grams of flour} \end{aligned}

Why multiply by 2.52.5? There are two and a half batches of the original four-serving recipe.

Answer. 750 g750\,\mathrm{g}.

Check. Each serving uses 300÷4=75 g300\div4=75\,\mathrm{g}, and 10×75=750 g10\times75=750\,\mathrm{g}.

Add one ingredient to the same recipe. It must use exactly the same multiplier.

Worked example 4: keeping two ingredients in balance

Question. The four-person recipe also uses 120 g120\,\mathrm{g} of butter. Work out the flour and butter needed for 1010 people. Non-calculator.

Plan. Multiply both ingredient masses by 2.52.5.

300×2.5=750grams of flour120×2.5=300grams of butter\begin{aligned} 300\times2.5&=750 &&\text{grams of flour}\\ 120\times2.5&=300 &&\text{grams of butter} \end{aligned}

Why use one factor for both? A different multiplier would change the recipe's flavour and texture.

Answer. 750 g750\,\mathrm{g} of flour and 300 g300\,\mathrm{g} of butter.

Check. 300:120=5:2300:120=5:2, and 750:300=5:2750:300=5:2.

You may instead work out a one-serving amount first. Both routes express the same scale change.

Watch out

Wrong: 10−4=610-4=6, so add 6 g6\,\mathrm{g} to every ingredient. Right: use a multiplier, 10÷4=2.510\div4=2.5, because a recipe is a multiplicative relationship.

This new recipe has two ingredients as well. Work with the given number of servings, not a guessed whole-number batch.

Mixtures and concentration

A mixture ratio compares the separate ingredients. Concentration describes how much of a chosen ingredient there is in a fixed amount of mixture.

For squash : water =1:4=1:4, there are five parts of drink altogether. This is a weaker drink than squash : water =1:3=1:3, because the same one part of squash is spread through more drink. For these questions assume that the volumes add without changing when mixed.

Worked example 5: making a mixture

Question. A drink uses squash : water =1:4=1:4. Work out the volumes needed for 750750 millilitres of drink. Non-calculator.

Plan. Share the total between five equal parts.

1+4=5total parts750÷5=150millilitres per part1×150=150squash in millilitres4×150=600water in millilitres\begin{aligned} 1+4&=5 &&\text{total parts}\\ 750\div5&=150 &&\text{millilitres per part}\\ 1\times150&=150 &&\text{squash in millilitres}\\ 4\times150&=600 &&\text{water in millilitres} \end{aligned}

Why divide by 55? The final drink includes both squash and water.

Answer. 150150 millilitres of squash and 600600 millilitres of water.

Check. 150+600=750150+600=750 and 150:600=1:4150:600=1:4.

Change one feature: add water after making the drink. Only the water amount increases.

Worked example 6: diluting a mixture

Question. Add 150150 millilitres of water to that 750750-millilitre drink. Work out the new squash : water ratio. Is the new drink stronger or weaker? Non-calculator.

Plan. Keep the squash amount unchanged, increase the water amount, then simplify.

600+150=750new water volume150:750=1:5new squash to water ratio\begin{aligned} 600+150&=750 &&\text{new water volume}\\ 150:750&=1:5 &&\text{new squash to water ratio} \end{aligned}

Why leave the squash at 150150? Adding water does not add squash.

Answer. 1:51:5; the drink is weaker because there is more water per part of squash.

Check. The total is now 900900 millilitres, with the original 150150 millilitres of squash.

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