C1.11; E1.11 - Scaling quantities and comparing value

C1.11; E1.11 - Scaling quantities and comparing value

Proportional reasoning connects a known quantity to a new one by keeping the same relative amounts. You will scale through one unit, adapt recipes, use map scales and compare prices fairly, while checking when proportional reasoning is justified.

Scaling through one unit

Five identical notebooks cost \12.Wouldeightcost. Would eight cost $15$ because both numbers have increased by three? Each extra notebook costs the same amount as every other notebook, not necessarily one dollar. Find that amount first.

Unitary method

The unitary method finds the amount for one unit, then scales to the required number of units. It applies when equal quantities have equal corresponding amounts: for example, identical notebooks sold at a fixed price each, with no extra charge or discount.

If you double the number of notebooks at a fixed unit price, you double the cost. This is a multiplicative connection. Adding the same number to both quantities usually does not preserve it.

Worked example 1: scaling up through one

Question. Five identical notebooks cost \12$. Work out the cost of eight at the same price each.

Plan. Find the price of one notebook, then multiply by eight.

12÷5=2.40dollars for one notebook2.40×8=19.20eight equal prices\begin{aligned} 12\div5&=2.40 &&\text{dollars for one notebook}\\ 2.40\times8&=19.20 &&\text{eight equal prices} \end{aligned}

Answer. \19.20$.

Check. The three extra notebooks cost 3\times\2.40=$7.20;; $12+$7.20=$19.20$.

Change only the required number from eight to three. Scaling down uses the same unit price.

Worked example 2: scaling down through one

Question. Five identical notebooks cost \12$. Work out the cost of three at the same price each.

Plan. Reuse the price for one unit, then take three units.

12÷5=2.40dollars per notebook2.40×3=7.20cost of three\begin{aligned} 12\div5&=2.40 &&\text{dollars per notebook}\\ 2.40\times3&=7.20 &&\text{cost of three} \end{aligned}

Answer. \7.20$.

Check. Three notebooks and two notebooks together cost \7.20+$4.80=$12$, recovering the five-notebook cost.

A fixed delivery charge breaks this simple model. If a total price includes \4$ delivery, doubling the order does not double that charge. Check the context before deciding that two quantities scale together.

Watch out

“Add three notebooks, so add three dollars” assumes a unit price of \1$ without evidence. A proportional change multiplies both quantities by the same factor.

Recipes and scale factors

A recipe preserves its proportions when every ingredient is multiplied by the same scale factor. Find that factor by dividing the required number of servings by the original number. It can be greater than 11, less than 11, or fractional.

Worked example 3: more servings

Question. A recipe for four people uses 240 g240\,\mathrm{g} of rice and 600 ml600\,\mathrm{ml} of stock. Find the amounts for ten people.

Plan. Multiply both ingredient amounts by 10÷410\div4.

10÷4=2.5scale factor for servings240×2.5=600grams of rice600×2.5=1500millilitres of stock\begin{aligned} 10\div4&=2.5 &&\text{scale factor for servings}\\ 240\times2.5&=600 &&\text{grams of rice}\\ 600\times2.5&=1500 &&\text{millilitres of stock} \end{aligned}

Answer. 600 g600\,\mathrm{g} rice and 1500 ml1500\,\mathrm{ml} stock.

Check. Rice per person is 600÷10=60 g600\div10=60\,\mathrm{g}, the same as 240÷4240\div4. Stock per person remains 150 ml150\,\mathrm{ml}.

Change only the required number of servings. All ingredients must still follow one common factor.

Worked example 4: a smaller scale factor

Question. Find the amounts of rice and stock for six people using the same four-person recipe.

Plan. Multiply the original amounts by 6÷46\div4.

6÷4=1.5new scale factor240×1.5=360grams of rice600×1.5=900millilitres of stock\begin{aligned} 6\div4&=1.5 &&\text{new scale factor}\\ 240\times1.5&=360 &&\text{grams of rice}\\ 600\times1.5&=900 &&\text{millilitres of stock} \end{aligned}

Answer. 360 g360\,\mathrm{g} rice and 900 ml900\,\mathrm{ml} stock.

Check. The rice-to-stock comparison is unchanged: 240:600=2:5240:600=2:5 and 360:900=2:5360:900=2:5.

A map uses the same principle for lengths. A scale of 1:50 0001:50\,000 means one unit on the map represents 50 00050\,000 of the same unit in reality. In particular, 1 cm1\,\mathrm{cm} represents 50 000 cm=500 m=0.5 km50\,000\,\mathrm{cm}=500\,\mathrm{m}=0.5\,\mathrm{km}.

Worked example 5: map to reality

Question. Two points are 3.6 cm3.6\,\mathrm{cm} apart on a map of scale 1:50 0001:50\,000. Find the actual distance in kilometres.

Plan. Use the actual distance represented by each centimetre.

1 cm on map⟶0.5 kmconvert the scale3.6×0.5=1.8scale the distance\begin{aligned} 1\,\mathrm{cm}\text{ on map}&\longrightarrow0.5\,\mathrm{km} &&\text{convert the scale}\\ 3.6\times0.5&=1.8 &&\text{scale the distance} \end{aligned}

Answer. 1.8 km1.8\,\mathrm{km}.

Check. 3.6×50 000=180 000 cm3.6\times50\,000=180\,000\,\mathrm{cm}, which is 1800 m=1.8 km1800\,\mathrm{m}=1.8\,\mathrm{km}.

Reverse only the direction of the question. Dividing undoes the map-to-reality multiplication.

Worked example 6: reality to map

Question. A road is 1.8 km1.8\,\mathrm{km} long. Find its length on a map of scale 1:50 0001:50\,000.

Plan. Find how many 0.5 km0.5\,\mathrm{km} lengths fit into 1.8 km1.8\,\mathrm{km}.

1.8÷0.5=3.6number of map centimetres\begin{aligned} 1.8\div0.5&=3.6 &&\text{number of map centimetres} \end{aligned}

Answer. 3.6 cm3.6\,\mathrm{cm}.

Check. 3.6×0.5=1.8 km3.6\times0.5=1.8\,\mathrm{km}, returning to the actual road length.

Watch out

The ratio 1:50 0001:50\,000 does not mean 1 cm1\,\mathrm{cm} represents 50 000 m50\,000\,\mathrm{m}. Its terms use the same unit. Convert the actual length only after interpreting the ratio.

Comparing value

The cheapest pack need not give the best value. Compare the cost of the same quantity, or the quantity bought for the same cost. Either approach is valid, but be consistent: a lower price per kilogram is better value; a greater mass per dollar is better value.

Worked example 7: price per kilogram

Question. Rice is sold in 750 g750\,\mathrm{g} packs for \2.40andand1.2,\mathrm{kg}packsforpacks for$3.60$. Which is better value?

Plan. Convert both masses to kilograms and compare price per kilogram.

750 g=0.75 kguse common units2.40÷0.75=3.20dollars per kg for the first pack3.60÷1.2=3.00dollars per kg for the second pack\begin{aligned} 750\,\mathrm{g}&=0.75\,\mathrm{kg} &&\text{use common units}\\ 2.40\div0.75&=3.20 &&\text{dollars per kg for the first pack}\\ 3.60\div1.2&=3.00 &&\text{dollars per kg for the second pack} \end{aligned}

Answer. The 1.2 kg1.2\,\mathrm{kg} pack is better value: \3.00perkilogramislowerthanper kilogram is lower than$3.20$.

Check. Six kilograms cost 8\times\2.40=$19.20insmallpacks,orin small packs, or5\times$3.60=$18.00$ in large packs.

Change only the large pack's price. Its larger size does not guarantee the conclusion.

Worked example 8: the larger pack loses

Question. The 750 g750\,\mathrm{g} pack still costs \2.40,butthe, but the 1.2,\mathrm{kg}packnowcostspack now costs$3.96$. Which is better value?

Plan. Compare the same unit prices again.

2.40÷0.75=3.20small pack: dollars per kg3.96÷1.2=3.30large pack: dollars per kg\begin{aligned} 2.40\div0.75&=3.20 &&\text{small pack: dollars per kg}\\ 3.96\div1.2&=3.30 &&\text{large pack: dollars per kg} \end{aligned}

Answer. The 750 g750\,\mathrm{g} pack is now better value.

Check. Six kilograms cost \19.20insmallpacks,comparedwithin small packs, compared with5\times$3.96=$19.80$ in large packs.

“Best value” here compares quantity for money, assuming the product quality is the same. The best purchase for a particular person can also depend on how much they need, whether they can buy partial packs, and whether unused product will be wasted.

Exam tip

Give comparable figures and a conclusion naming the pack. Comparing the two total prices alone does not establish value when pack sizes differ.

Quick Recap

  • For a proportional relationship, scale both quantities by the same factor.
  • The unitary method goes through one unit; recipe scale factors multiply every ingredient.
  • Map ratios compare lengths in the same units before conversion.
  • Compare equal quantities or equal costs to decide best value.

Eight notebooks in the opening question cost \19.20,becauseeachcosts, because each costs $2.40$. Adding three to both given numbers would not preserve that price.