1.7 Ratio and proportion

Learning objectives

1.7A Ratio notationuse ratio notation, including reduction to its simplest form and its various links to fraction notation Express in the form 1 : n 1.7B Divide a quantity in a given ratio or ratios Share £416divide a quantity in a given ratio or ratios Share £416 in the ratio 5 : 3 or 4 : 3 : 1 1.7C The process of proportionality to evaluate unknownuse the process of proportionality to evaluate unknown quantities 1.7D Missing quantities in direct proportionCalculate an unknown quantity when quantities vary in direct proportion, including completing a value table.1.7E Word problems about ratio and proportionsolve word problems about ratio and proportion Including maps and scale diagrams

A ratio compares quantities multiplicatively and in a stated order. The ratio $a:b$ means that for every $a$ parts of the first quantity there are $b$ parts of the second.

Job	Example
write in order	32 teeth to 24 teeth is $32:24$
simplify	$32:24=4:3$
express as $1:n$	$5:8=1:1.6$
link to a fraction	in $4:3$ , the first share is $4/(4+3)=4/7$ of the total

Convert quantities to the same units before forming or simplifying a ratio: $2 ext{ m}:50 ext{ cm}=200:50=4:1$ .

Multiplying or dividing every part by the same non-zero factor gives an equivalent ratio.

Order matters: $4:3$ and $3:4$ compare opposite directions. Do not simplify by subtracting the same number from both parts.

To share a total in a ratio, treat the ratio numbers as equal-sized part counts.

Step	Share £416 in $5:3$
add ratio parts	$5+3=8$ parts
find one part	$416\div8=52$
multiply for each share	$5 imes52=260$ , $3 imes52=156$

For a three-part ratio such as $4:3:1$ , add all three parts before finding the value of one part.

Check that the shares total the original quantity and that their simplified ratio matches the given ratio.

Do not divide the total separately by each ratio number. The denominator is the sum of all ratio parts.

Directly proportional quantities change by the same scale factor. If one quantity is multiplied by $k$ , the linked quantity is also multiplied by $k$ .

Information	Proportional step
8 calculators cost £62.80	one costs $62.80\div8=7.85$
cost of 12 calculators	$7.85 imes12=£94.20$
equivalent scale-factor method	$62.80 imes(12/8)=£94.20$

Use a unitary method when one-unit value is useful, or move directly by a scale factor when the relationship is clear.

For direct proportion, the ratio $y/x$ is constant, so the model can be written $y=kx$ .

Adding the same difference is not proportional scaling. The relationship must pass through zero: zero items cost zero at a fixed unit price.

A direct-proportion table contains pairs linked by one constant multiplier. Find that multiplier or reduce to one unit, then complete every missing entry.

Cakes	Flour
15	360 g
1	$360\div15=24$ g
38	$38 imes24=912$ g

If $y$ varies directly as $x$ , write $y=kx$ . Use one known pair to calculate $k$ , then substitute the required value.

Keep corresponding columns in consistent units. For example, convert $0.85 ext{ kg}$ to $850 ext{ g}$ before comparing it with $912 ext{ g}$ .

Do not assume a table is proportional merely because both columns increase. Verify that $y/x$ stays constant.

A word problem must first be translated into the comparison being held constant: ratio parts, a unit rate, a scale factor or a map scale.

Context	Useful comparison
best value	cost per unit or amount per currency unit
recipe	amount per serving, then scale
map	actual distance $=$ map distance $imes$ scale
mixture	total parts, then each ingredient's fraction

A 150 g bag costing £1.80 costs $1.80/150=£0.012$ per gram. A 400 g bag costing £5 costs $5/400=£0.0125$ per gram, so the small bag is better value.

Compare like with like, show the common unit rate or equivalent quantity, and state the conclusion in context.

A larger pack is not automatically better value. Its price and quantity must be compared using the same unit.

Next Topic

—1.8 Degree of accuracyRead

1.7 Ratio and proportion

Learning objectives

Write and simplify ratios

Divide a quantity in a given ratio

Use proportionality to find unknown quantities

Complete direct-proportion quantities and tables

Solve ratio and proportion word problems

Next Topic