Alliance Form 3 P1 Q24 — Commission & Percentages (KCSE Maths)

KCSE (Form 3) Form 3 Numbers

Published

The Question

“A saleswoman earns a 2% commission on any sales above 100,000 shillings, plus a fixed monthly salary of 12,000 shillings. (a) In one month she sold 360 handbags at 500 shillings each. Calculate her total earnings for that month. (b) The following month her salary was increased by 10% and her total earnings were 17,600 shillings. Find the total value of her sales that month and the number of handbags she sold, given each handbag still sold at 500 shillings.”

1

Find this month's sales value

Start by turning the handbags into money. She sold 360 handbags and each one cost 500 shillings, so multiply the two together to get the total value of what she sold this month.

360×500=180,000 shillings360 \times 500 = 180{,}000 \text{ shillings}
2

Work out the commission

Commission is only paid on the part of the sales that is above 100,000 shillings, not on the whole amount. Subtract the 100,000 threshold first, then take 2% of what is left.

180,000−100,000=80,000180{,}000 - 100{,}000 = 80{,}000
2%×80,000=2100×80,000=1,600 shillings2\% \times 80{,}000 = \frac{2}{100} \times 80{,}000 = 1{,}600 \text{ shillings}
3

Add the salary to get part (a)

Her total pay is the commission she just earned plus her fixed monthly salary of 12,000 shillings. Add them to finish part (a).

1,600+12,000=13,600 shillings1{,}600 + 12{,}000 = 13{,}600 \text{ shillings}
4

Find the new salary for part (b)

The next month her salary rises by 10%. Increase the original 12,000 by 10% of itself, which means multiplying by 1.1, to get her new salary.

12,000×1.1=13,200 shillings12{,}000 \times 1.1 = 13{,}200 \text{ shillings}
5

Separate out the new commission

We are told her total earnings that month were 17,600 shillings. Since earnings are salary plus commission, subtract the new salary to leave only the commission she must have earned.

17,600−13,200=4,400 shillings17{,}600 - 13{,}200 = 4{,}400 \text{ shillings}
6

Work backwards to the total sales

That 4,400 is 2% of the sales above 100,000. To reverse a 2% rate, divide by 2 and multiply by 100, which is the same as multiplying by 50. Then add the 100,000 threshold back on to recover the full sales figure.

4,400÷2100=4,400×50=220,0004{,}400 \div \frac{2}{100} = 4{,}400 \times 50 = 220{,}000
220,000+100,000=320,000 shillings220{,}000 + 100{,}000 = 320{,}000 \text{ shillings}
7

Convert sales back into handbags

Each handbag still sells at 500 shillings, so divide the total sales value by the price per handbag to find how many she sold.

320,000÷500=640 handbags320{,}000 \div 500 = 640 \text{ handbags}

Final Result

(a) Her earnings for the first month are 13,600 shillings. (b) The following month her total sales were 320,000 shillings, which is 640 handbags.

Why this method works

The method works because earnings are built from two separate pieces — a fixed salary that never changes with sales, and a commission that depends only on the sales above a threshold. Going forwards you build earnings up: money from sales, cut off the first 100,000, take 2%, and add the salary. Going backwards you undo each move in reverse order: strip off the salary to expose the commission, reverse the 2% to recover the sales above the threshold, add the threshold back, and finally divide by the unit price. Because every operation has an exact inverse, the same steps run cleanly in either direction.

✓

Check part (b): 2% of (320,000 - 100,000) = 0.02 x 220,000 = 4,400 commission; add the salary 13,200 to get 17,600 total earnings, which matches the given figure.