10.1 Ratios, Percentages and Modelling
Key Takeaways
- MAT ratio items require an algebraic model; QL ratio items usually ask you to read a table or tariff and compute a comparison.
- A p% change is a multiplier of 1 + p/100 or 1 − p/100; successive changes multiply and do not add.
- A part-to-part ratio a:b splits a whole into a/(a+b) and b/(a+b); mixing that with part-to-whole is a standard MAT distractor.
- Reverse a percentage change by dividing by the multiplier, not by applying the same percentage in the opposite direction.
- On the calculator-free MAT paper, rewrite percentages as fractions before you solve, then match an option.
The NBT Mathematics (MAT) paper is a separate three-hour multiple-choice test from the AQL paper. Current nbt.ac.za MAT Test Content includes applying ratios and percentages in a variety of contexts, and solving problems using mathematical process skills — in particular, translating language into algebra. That demand is not the same as Quantitative Literacy (QL). A QL item typically asks you to read a table, a campus tariff, or a news-style percentage and compute a numerical comparison. A MAT item expects you to introduce a variable, write an equation that captures the ratio or percentage relationship, and reason to an exact answer with no calculator.
This OpenExamPrep chapter is independent exam-prep teaching. It uses original numbers and original stems. It is not an NBTP or CEA booklet and it does not copy official exemplar items.
What modelling a ratio looks like on MAT
A ratio a:b is a part-to-part split of one whole when the two quantities together make that whole. The first quantity is then a/(a+b) of the whole and the second is b/(a+b). Writers who treat a:b as a/a and b/a, or who mix part:part with part:whole, lose the item even when the arithmetic is easy.
Worked sharing model. Three co-founders split a seed grant in the ratio 4:5:7. The largest share exceeds the smallest share by R9 600. Find the total grant.
Let the parts be 4k, 5k and 7k rand. The sentence "largest minus smallest is 9 600" becomes 7k − 4k = 9600, so 3k = 9600 and k = 3200. The total is 16k = R51 200.
The usual MAT trap is to pair the wrong two parts. Subtracting the middle from the largest, 7k − 5k = 9600, gives k = 4800 and a total of R76 800 — a plausible wrong option. The process skill is: name every part with one parameter, then write the constraint as an equation before you compute.
Percentages are operators, not labels you add
A p% increase multiplies by (1 + p/100). A p% decrease multiplies by (1 − p/100). Successive changes multiply; they do not add. That single fact separates a MAT modelling item from a Grade 9 arithmetic drill.
Worked successive-change model. A listed price of R640 is increased by 25% for a holiday surcharge, then reduced by 20% in a sale. What is paid?
The first multiplier is 5/4. The second is 4/5. The product is 1, so the amount paid is still R640. The 25% and the 20% do not "cancel" because the numbers look similar; they cancel because (5/4)×(4/5) = 1. A 25% increase followed by a 25% decrease does not return to the start: (5/4)×(3/4) = 15/16, a net fall of 1/16, which is 6.25%.
The bar chart after this section shows the same idea on a base of 100: up 20% then down 20% lands on 96, not back on 100. If an option says "no net change" for equal-looking percentages in opposite directions, test the multipliers before you agree.
Concentration and the conserved quantity
Worked dilution model. A 36 litre tank of cleaning solution is 20% concentrate. How much water must be added so that the mixture is 12% concentrate?
The concentrate volume is fixed: (1/5)×36 = 36/5 litres. After adding w litres of water,
(36/5) / (36 + w) = 12/100 = 3/25.
Cross-multiply: (36/5)×(25/3) = 36 + w. That simplifies to 12×5 = 60 = 36 + w, so w = 24 litres.
On MAT, rewriting percentages as fractions is usually faster than working in decimals, and it matches the no-calculator rule. QL might instead show a lab table of concentrations and ask which sample is more dilute. MAT asks you to write the conserved-quantity equation.
Translating "of", "more than", and "is to"
Language-to-algebra is a listed MAT process skill. Keep a short translation table and use it on every stem.
| English in the stem | Algebra |
|---|---|
| A is 30% of B | A = (3/10)B |
| A is 30% more than B | A = (13/10)B |
| A is 30% less than B | A = (7/10)B |
| A is to B as 3 is to 5 | A/B = 3/5 |
| A exceeds B by 30% of B | A − B = (3/10)B |
| Ratio A:B = 3:5 and A + B = 48 | A = 18, B = 30 |
Worked reverse-percentage model. After a 15% pay cut, a stipend is R2 040 per month. What was the stipend before the cut?
The current amount is 85% of the original: (17/20)P = 2040. Then P = 2040 × 20/17. First 2040 ÷ 17 = 120, then 120 × 20 = 2400. The original stipend was R2 400.
The popular error is to increase R2 040 by 15%, which is 2040 × 23/20 = 2346. That is not the reverse of a 15% decrease. Reverse a multiplicative change by dividing by the multiplier.
A two-mixture model
Mixture A is copper and zinc in the ratio 7:3. Mixture B is copper and zinc in the ratio 1:1. Combine 40 kg of A with x kg of B so that the result is copper:zinc = 3:2. Find x.
From A, copper is (7/10)×40 = 28 kg and zinc is 12 kg. From B, each metal is x/2 kg. Then
(28 + x/2) : (12 + x/2) = 3 : 2,
so 2(28 + x/2) = 3(12 + x/2). That is 56 + x = 36 + 3x/2. Then 20 = 3x/2 − x = x/2, so x = 40 kg.
The target ratio 3:2 is not a percentage you punch into a display. It is a proportion that becomes a linear equation in x. QL might ask what percentage of a finished blend is copper after both masses are given. MAT asks you to find the unknown mass that forces the target ratio.
Inverse ratio and rates
If 6 printers of equal speed produce a run in 8 hours, how many printers finish the same run in 6 hours? Printer-hours are constant: 6 × 8 = 48, so n × 6 = 48 and n = 8. That is inverse proportion: (number of printers) × (time) = constant.
A MAT stem may hide the same structure in words about filling a reservoir. Inlet A alone fills a tank in 12 hours; inlet B alone fills it in 18 hours. In one hour they fill 1/12 + 1/18 = 5/36 of the tank, so together they take 36/5 hours, which is 7 1/5 hours. Keep the answer as a fraction; MAT will typically offer 36/5 among the options rather than a decimal.
Process checklist before you open the options
- Underline the relationship words: of, more than, less than, to, per, in the ratio.
- Name one variable (sometimes two, then eliminate).
- Write a single equation that uses every constraint.
- Solve with fractions, not with estimated decimals.
- Only then look at the four options.
On the MAT afternoon paper you have no calculator and no scaffolding. The ratio item will not say "first find k, hence find the total." It will ask for the total, or for x, and the four options will include the arithmetic of the most common mis-model. Work the algebra before matching; use only the working surface permitted for your test mode.
Before you move on
- QL reads a percentage in a table; MAT writes a percentage as a multiplier or an equation.
- Successive percentage changes multiply. Equal-looking up-and-down percentages rarely return the original amount.
- Conserved quantities (salt, acid, copper) stay in the numerator while the whole changes.
- Inverse problems keep a product constant. Rate-of-work problems add reciprocals of times.
A 40 litre solution is 15% acid. Water is added until the mixture is 10% acid. How many litres of water are added?
Three partners share a profit in the ratio 2:3:5. The middle share is R2 700 more than the smallest share. The total profit is
A number is increased by 25% and the result is then decreased by 20%. Relative to the original, the final value is