KCSE 2025 Maths P2 Q17 — Mixtures, Percentage Profit and Ratio

KCSE 2025 Form 4 Commercial Arithmetic

Published

The Question

“Two varieties of groundnuts are sold in the market. Type Q sells at Ksh 130 per kg and type R sells at Ksh 180 per kg. (a)(i) A trader buys 50 kg of type Q and 75 kg of type R and mixes them; find the cost price of 1 kg of the mixture. (a)(ii) The trader sells 80% of the mixture at Ksh 170 per kg and the rest at Ksh 180 per kg; find the percentage profit made. (b) A different trader mixes the two varieties and sells the mixture at Ksh 200 per kg, making a profit of 25%; find the ratio in which the two varieties were mixed.”

1

Find the cost of 1 kg of the mixture (a)(i)

The cost price of one kilogram of a mixture is the total money spent divided by the total mass. Work out what each variety cost, add the two to get the total cost of the batch, then add the two masses to get the total mass. Dividing the total cost by the total mass gives the cost per kilogram.

Total cost=50×130+75×180=6 500+13 500=20 000\text{Total cost} = 50 \times 130 + 75 \times 180 = 6\,500 + 13\,500 = 20\,000
Total mass=50+75=125 kg\text{Total mass} = 50 + 75 = 125 \text{ kg}
Cost per kg=20 000125=160\text{Cost per kg} = \frac{20\,000}{125} = 160

Keep the Ksh 160 per kg figure — it is used again in part (b).

2

Split the mixture and find the revenue (a)(ii)

The trader sells the 125 kg in two portions, so first work out each portion's mass: 80% and 20% of 125 kg. Then find the revenue from each portion by multiplying its mass by its selling price, and add them to get the total money brought in.

80%×125=100 kg,20%×125=25 kg80\% \times 125 = 100 \text{ kg}, \qquad 20\% \times 125 = 25 \text{ kg}
Revenue=100×170+25×180=17 000+4 500=21 500\text{Revenue} = 100 \times 170 + 25 \times 180 = 17\,000 + 4\,500 = 21\,500
3

Work out the percentage profit (a)(ii)

Profit is the revenue minus the cost of the whole batch. The batch cost Ksh 20,000 from part (i), so subtract that from the revenue. The percentage profit is this profit written as a fraction of the cost, multiplied by 100.

Profit=21 500−20 000=1 500\text{Profit} = 21\,500 - 20\,000 = 1\,500
% profit=1 50020 000×100=7.5%\%\,\text{profit} = \frac{1\,500}{20\,000} \times 100 = 7.5\%
4

Find the mixture cost for the second trader (b)

When a 25% profit is made, the selling price is 125% of the cost, i.e. 1.25 times the cost. Since the second trader sells at Ksh 200 per kg, divide the selling price by 1.25 to recover the cost price of one kilogram of that trader's mixture.

200=1.25×cost⇒cost=2001.25=160200 = 1.25 \times \text{cost} \Rightarrow \text{cost} = \frac{200}{1.25} = 160
5

Set up and solve for the mixing ratio (b)

Let the trader mix Q and R in the ratio m to n. The cost per kilogram of the mixture is the weighted average of the two prices, and this must equal the Ksh 160 just found. Cross-multiply, collect the m terms on one side and the n terms on the other, then simplify to get the ratio.

130m+180nm+n=160\frac{130m + 180n}{m + n} = 160
130m+180n=160(m+n)=160m+160n130m + 180n = 160(m + n) = 160m + 160n
180n−160n=160m−130m⇒20n=30m180n - 160n = 160m - 130m \Rightarrow 20n = 30m
mn=2030=23\frac{m}{n} = \frac{20}{30} = \frac{2}{3}

Final Result

The cost price of 1 kg of the mixture is Ksh 160. The percentage profit made by the first trader is 7.5%. The second trader mixed type Q and type R in the ratio 2 : 3.

Why this method works

Cost per kilogram is a rate — total shillings shared over total kilograms — so both the money and the mass must be pooled before dividing; that same weighted-average idea drives part (b), where the mixture's cost is the price of each variety weighted by how much of it is present. Percentage profit compares gain against outlay, so the profit is measured relative to the Ksh 20,000 actually spent, not the revenue. In part (b), a 25% markup means selling price is a fixed multiple (1.25) of cost, which is why dividing by 1.25 unwinds the markup to reveal the true cost the ratio must produce.

✓

In the ratio 2 : 3, take 2 kg of Q and 3 kg of R: cost = (2 x 130 + 3 x 180) / 5 = (260 + 540) / 5 = 800 / 5 = 160 per kg, matching the required cost.