Learning Objectives
Translate a ratio into fractions of a total.
Combine successive percentage changes using multipliers.
Recover an original value from a changed value.
Build direct and inverse proportion models with stated assumptions.
Prerequisites: exact fractions and index laws from the preceding two lessons. All practice questions in this lesson are original, not official TMUA questions.
What This Lesson Is About
Ratios, percentages and proportion all describe multiplicative relationships, but they answer different questions. A ratio compares quantities, a percentage expresses a quantity relative to a reference amount, and a proportional model describes how one quantity changes with another.
This lesson starts by identifying the reference whole. It then turns percentage changes into multipliers, builds equations for direct and inverse proportion, and applies those ideas to mixtures and similar shapes. The aim is to choose the correct relationship before doing the arithmetic.
1. Ratios: Comparing Parts and Wholes
A statement such as A:B = 3:5 tells you how the two parts compare, but it does not give either quantity until you know their common scale. Start by deciding whether the question compares one part with another or one part with the total.
1.1 Part-to-part and part-to-whole comparisons
A ratio compares quantities using a common unit. If A:B = 3:5, write A = 3t and B = 5t for some positive scale t when both quantities are positive. The fraction A/B is 3/5, but A/(A + B) is 3/8. A ratio of parts is different from a fraction of the total.
There are 3 + 5 = 8 equal-sized parts in the total. This is why the share belonging to A uses denominator 8, whereas comparing A directly with B uses denominator 5.
1.2 Linked ratios: matching the shared quantity
If two ratios both mention B, their displayed B parts may use different scales. Rescale them so that B has the same number of parts in each before combining the relationships.
To combine two ratios, make the shared quantity represent the same number of parts. This is equivalent to finding a common denominator in fraction arithmetic. State whether the quantities are masses, volumes, counts or lengths before combining them.
Worked Example: Link Two Ratios
Question: A:B = 2:3 and B:C = 4:5. Find A:B:C and the share of A in the total.
Method: Rescale the two ratios until the common quantity B has the same number of parts, then count the total parts.
The shared B must match. Multiply the first ratio by 4 and the second by 3: A:B = 8:12 and B:C = 12:15. Thus A:B:C = 8:12:15.
There are 35 parts in total, so A is 8/35 of the total. Check: 8/12 = 2/3 and 12/15 = 4/5.
2. Percentage Changes: Multiplying and Reversing
A percentage change is measured relative to a particular starting amount. Turning it into a multiplier keeps that reference explicit and gives a reliable way to combine changes or work backwards to an original value.
2.1 A percentage multiplier: retaining the original and adding the change
p% means p/100. Increasing a positive quantity by p% multiplies it by 1 + p/100. Decreasing it by p% multiplies it by 1 − p/100. For an ordinary reduction that leaves a nonnegative amount, use 0 ≤ p ≤ 100.
For example, a 20% increase keeps 100% of the original and adds 20%, leaving 120% = 1.2 times the original. A 20% reduction keeps 80% = 0.8 times the original.
2.2 Successive changes and changes in a percentage
Successive changes act on different reference amounts. Multiply their factors rather than adding the percentages. A percentage point, by contrast, describes a difference between two percentages: a rise from 20% to 25% is 5 percentage points, or a 25% increase relative to the original 20%.
In a sequence of changes, the output of the first step becomes the input of the next. When comparing two percentages themselves, name whether you mean their numerical difference in percentage points or a relative increase. Those are different measurements.
2.3 Reversing a change: divide by its multiplier
If final value = multiplier × original value, dividing the final value by the nonzero multiplier recovers the original. This explains why undoing a discount does not mean adding the same percentage to the reduced price.
Worked Example: A Rise and a Fall
Question: A price rises by 20% and then falls by 20%. What is the overall change?
Method: Write a multiplier for each change and multiply them, keeping the second change relative to the changed price.
The combined multiplier is (6/5)(4/5) = 24/25 = 0.96. The final price is 96% of the original, a 4% decrease.
Check with an original price of £100: £100 becomes £120, then £96. The second change removes £24 because it acts on £120.
Worked Example: Recover the Original
Question: After a 15% discount, an item costs £68. Find its original price.
Method: The discounted price is 85% of the original, so divide by that remaining fraction to recover the starting price.
Let the original be P. Then (17/20)P = 68, so P = 68 × 20/17 = £80.
Check: 15% of £80 is £12, and £80 − £12 = £68. Adding 15% to £68 uses the wrong reference amount.
3. Proportion: Describing How Quantities Vary
A proportional model gives a rule for a whole family of corresponding values, not just one comparison. Identify the quantity that remains constant, write an equation, and then use it to predict a change.
3.1 Direct and inverse proportion: ratio or product stays fixed
Direct proportion y = kx means y/x is constant for nonzero x. The constant k is determined by one pair of corresponding values. Inverse proportion y = k/x means xy is constant, with x ≠ 0. Doubling x doubles y in the first model and halves y in the second.
3.2 Powers in a proportional model: scaling by more than x
Some relationships depend on an area-like square, a root, or another power of the input. In those cases, doubling the input does not necessarily double or halve the output. Apply the specified power to the input’s scale factor.
More generally, y may be proportional to a power of x. If y = kx², multiplying x by a factor s multiplies y by s². If y = k/√x with x > 0, multiplying x by s > 0 multiplies y by 1/√s. These statements follow by substituting sx into the model.
3.3 Checking the model: increasing is not the same as proportional
A proportional model is an assumption, not a consequence of two quantities increasing together. For example, y = 3x + 2 is increasing but not directly proportional to x because y/x is not constant.
A nonzero fixed addition changes the ratio as the input changes. Before using a proportional rule in a context, make sure the relationship is given or justified rather than inferred just from an upward trend.
Worked Example: Scale without Finding the Constant
Question: A positive quantity y is inversely proportional to √x. If x changes from 9 to 25, by what factor does y change?
Method: Write the old and new values using the same proportionality constant and compare them as a ratio.
Write y = k/√x. The original value is k/3 and the new value is k/5. Their ratio is (k/5)/(k/3) = 3/5.
Thus y decreases by 40%. Check using any positive k, such as 15: y changes from 5 to 3.
4. Applying Proportion to Mixtures and Similar Shapes
The previous models can now be used in two settings. In a mixture, the key is to keep track of each ingredient’s share of a total. In similar shapes, the key is to decide whether the quantity being compared is a length, area or volume.
4.1 Mixture concentration: add ingredient masses before finding the share
Mass percentage means the ingredient’s mass divided by the total mixture mass, multiplied by 100. When combining portions, first find the actual ingredient mass contributed by each portion; the percentages alone do not tell you those contributions.
For a mixture described by mass percentage, calculate the mass of the ingredient in each portion and add these masses. Divide by the total mass afterwards. Averaging the percentages directly works only when the portion masses are equal.
4.2 Similarity: length, area and volume scale differently
Similar shapes have matching angles and corresponding lengths in one common ratio. If every length is multiplied by s, two length factors occur in an area and three in a volume, giving different powers of s.
For similar shapes with positive length scale factor s, areas scale by s² and volumes by s³. The shapes must be similar: doubling just one dimension does not generally multiply volume by eight.
Worked Example: A Weighted Mixture
Question: Mix 200 g of a 15% salt solution with 300 g of a 5% salt solution. Assume no material is lost. Find the final mass percentage of salt.
Method: Find the salt mass in each portion, add the masses, and divide by the total solution mass.
Salt masses are (15/100) × 200 = 30 g and (5/100) × 300 = 15 g. Total salt is 45 g in 500 g of solution.
The mass fraction is 45/500 = 9/100, so the final concentration is 9%. It is nearer 5% than 15% because more of the weaker solution was used.
Common Mistakes
Keep part-to-part ratios separate from fractions of the whole.
Percentage changes use the current amount at each step, so equal increases and decreases do not cancel.
Use the model equation to decide what is constant; inverse proportion is not a constant difference.
Do not average mixture percentages unless the portion sizes are equal.
5. Practice
Start with ratios and single changes, then move to models, similarity and mixtures. In each question, state the reference amount or constant relationship before calculating. The last question checks whether you can reverse a percentage reduction exactly.
1. Divide 84 in the ratio 2:5.
2. A value increases by 25% and then decreases by 10%. Find the overall percentage change.
3. A price including a 20% increase is £90. Find the original price.
4. y is directly proportional to x² and y = 18 when x = 3. Find y when x = 5.
5. Two similar solids have volume ratio 8:27. Find their length ratio and surface-area ratio.
6. Mix 100 g of a 20% solution with 400 g of a 5% solution of the same substance. Find the final mass percentage, assuming no loss.
7. A quantity is reduced by 30%. What percentage increase on the reduced value restores the original? Give an exact answer.
Worked Solutions
1. Divide the total into equal ratio parts
There are 7 parts; each is 84/7 = 12. The quantities are 24 and 60. They sum to 84 and 24/60 = 2/5.
2. Multiply successive percentage factors
The multiplier is (5/4)(9/10) = 9/8. The net increase is 1/8 = 12.5%. Starting at 80 gives 100 then 90, a rise of 10 on 80.
3. Reverse the percentage multiplier
The original is 90/(6/5) = £75. Check: £75 + £15 = £90.
4. Find and use the proportionality constant
y = kx² and 18 = 9k, so k = 2. At x = 5, y = 50. The value ratio 50/18 = 25/9 matches the squared input ratio.
5. Convert a volume ratio to length and area ratios
The positive cube roots give length ratio 2:3. Squaring gives surface-area ratio 4:9. This uses similarity, not just equal volumes.
6. Weight the mixture by portion mass
Ingredient masses are 20 g and 20 g. The fraction is 40/500 = 2/25, hence 8%. The answer lies between 5% and 20%.
7. Find the inverse multiplier for restoration
The remaining multiplier is 7/10. Restoration requires multiplying by 10/7, an increase of 3/7 of the reduced value. The exact increase is (300/7)%. Check: (7/10)(10/7) = 1.
6. Summary
- Define the reference whole before turning a ratio into a fraction.
- Multiply percentage factors and divide by a factor to reverse a change.
- Translate proportionality into an equation before scaling.
- Use weighted totals for mixtures and powers of the length factor for similar shapes.