Word problems about sharing, mixing, working together, or travelling all reduce to one idea: how a change in one quantity moves another. Candidates who cannot tell direct proportion from inverse proportion will produce a confident, wrong answer in the same number of seconds as the right one — which is why this distinction is tested so often.
The second half of the value here is arithmetic hygiene. Ratio questions hide a step — turning a ratio into parts — and skipping it is how 3-part questions become 2-part mistakes.
A ratio is a recipe. The ratio 2 : 3 : 4 does not say the quantities are 2, 3 and 4; it says they are 2 parts, 3 parts and 4 parts. Convert to real quantities in two steps:
Proportion is the statement that two ratios are equal, and it answers "if this scales, what does that become". The only decision is the direction of the scaling:
Inverse proportion is where most errors happen, so make the check explicit: if the answer you computed moves in the same direction as the input you changed, you have used the wrong rule.
Splitting a quantity. Divide 4500 in the ratio 2 : 3 : 4. The terms total 9 parts, so one part is 4500 / 9 = 500. The shares are 1000, 1500 and 2000. Always verify: 1000 + 1500 + 2000 = 4500, and the shares keep the ratio when divided by 500.
Simplifying. Divide both terms by their highest common factor. 24 : 36 becomes 2 : 3. Keep ratios in whole numbers where you can; decimals in a ratio usually mean a step was missed.
Direct proportion. Set up a proportion and cross-multiply. If 3 identical pens cost 120, what do 7 cost? One pen is 40, so 7 cost 280. The cross-multiplied check: 3 / 120 = 7 / 280, both equal 0.025.
Inverse proportion. Use the product, not the ratio. If 12 workers build a wall in 10 days, the job is 12 x 10 = 120 worker-days. With 15 workers it takes 120 / 15 = 8 days. The sanity check is the direction: more workers, fewer days.
Rate problems. A rate is a ratio with different units. At 60 kilometres per hour, a distance of 120 kilometres takes 2 hours, and 90 minutes is 1.5 hours, so the same rate covers 90 kilometres. Convert time to hours before multiplying — the most common slip here is treating 90 minutes as 0.9 hours.
Mixtures. Track the amount of the substance, not the percentage. Forty litres of a solution that is 20 percent salt contains 20 percent of 40, that is 8 litres of salt. Add 10 litres of water and the solution is 50 litres, but the salt is still 8 litres, so it is now 8 / 50 = 16 percent.
Combining two mixtures is a weighted average. Mix 2 parts of a 30 percent solution with 3 parts of a 20 percent solution: the salt is 2 × 30 + 3 × 20 = 120 parts against 5 parts of solution, giving 120 / 5 = 24 percent.
A full mixture, step by step. A 60-litre mixture is 40 percent acid. How much water must be added to turn it into a 25 percent solution? Track acid, never the percentage. Acid now: 40 percent of 60 = 24 litres. Water does not change the acid, so after adding w litres the acid is still 24 and the total is 60 + w. Set 24 / (60 + w) = 0.25, so 60 + w = 96 and w = 36 litres. Check: 24 / 96 = 0.25. The same layout works when a stronger solution is mixed in instead of water — only the acid term changes.
A map uses a scale of 1 to 50,000. Two towns are 3 centimetres apart on the map. What is the real distance in kilometres?
Step 1, the direction: a bigger distance on the map means a bigger real distance, so this is direct proportion. One centimetre on the map represents 50,000 centimetres in reality, so 3 centimetres represent 150,000 centimetres.
Step 2, the units: 150,000 centimetres is 1,500 metres, which is 1.5 kilometres. The unit conversion is where most candidates lose the mark, not the proportion.
Now the same question as an inverse case. If the map scale were 1 to 100,000 and the real distance were 1.5 kilometres, the map distance would be smaller: 150,000 centimetres in reality divided by 100,000 gives 1.5 centimetres on the map. Same three numbers, opposite direction, and the sanity check — a larger scale number means a smaller map distance — catches the error before the answer is written down.
Divide 4500 in the ratio 2 : 3 : 4.
If 12 workers build a wall in 10 days, how long will 15 workers take?
Forty litres of a solution is 20 percent salt. After adding 10 litres of water, what is the salt concentration?
Without looking back: what is the difference between direct and inverse proportion, and which one applies when more workers join a fixed job?
The answer is: in direct proportion both quantities grow together; in inverse proportion one grows as the other falls. A fixed job is inverse proportion, so more workers means fewer days.
Fill in the blanks from memory:
Dividing 4500 in the ratio 2 : 3 : 4 means the parts total 9, so one part is . If 12 workers take 10 days, the job is worker-days, so 15 workers take 8 days. A solution of 40 litres that is 20 percent salt contains 8 litres of salt, and after adding 10 litres of water it is percent salt.
A recipe for 4 people uses 300 grams of rice and 200 grams of lentils. You must cook for 10 people and you want to keep the ratio. You have 800 grams of rice and 450 grams of lentils. What is the minimum you can cook for, and which ingredient limits you?
Scale the recipe by 10 / 4 = 2.5: the exact amounts needed are 750 grams of rice and 500 grams of lentils. You have enough rice (800 is more than 750) but not enough lentils (450 is less than 500). So the limiting ingredient is the lentils.
To find how many people you can actually cater for, invert the ratio from the lentils: 450 grams supports 450 / 200 = 2.25 times the original recipe, which is 2.25 x 4 = 9 people. Check the rice at that scale: 2.25 x 300 = 675 grams, which fits inside the 800 available, so the lentils remain the limit.
The reasoning pattern to keep — scale by a factor, compare both ingredients, then identify the binding constraint — is the same pattern used in every ratio-and-capacity question, whatever the units.
Ratios compare quantities that are already known. The next lesson handles the summary of a group: the average, the weighted average, and the mixture. It is where ratio reasoning pays off directly, because a weighted mean is just a ratio in disguise.