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Commercial arithmetic · Lesson 1 of 3

Percentages, profit and loss

Percentage change as a multiplier, profit and loss on the cost price, marked price and discount, working back to the original price, and successive percentage changes.

16 minYou should already know: Number foundations & fractions
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A percentage is a number out of 100: 35%35\% means 35100\frac{35}{100}, or 0.350.35. Almost every money question in this lesson comes down to one idea: a percentage change is a multiplication.

Percentage change is a multiplier

Increasing an amount by r%r\% gives (100+r)%(100 + r)\% of it, so multiply by 1+r1001 + \frac{r}{100}. Decreasing by r%r\% leaves (100−r)%(100 - r)\%, so multiply by 1−r1001 - \frac{r}{100}.

  • A 20%20\% increase is ×1.2\times 1.2.
  • A 15%15\% decrease is ×0.85\times 0.85.
100%original100%20%× 1.285%× 0.85
Percentage changeIncrease: × (1 + r/100). Decrease: × (1 − r/100)
Percentage change as a multiplierChange the amount and the percentage
100% = 500120% = 600each grid line is 10% of the original
× 1.2multiplier600500 increased by 20%
Increasing by 20% leaves 120% of the original, so multiply by 1.2: 500 × 1.2 = 600.

Try Find the original. The new value is the original times the multiplier, so to go back you divide by the multiplier. Then try Two reductions: the two changes multiply, they don’t simply add.

Profit and loss

Profit or loss is always a percentage of the cost price (what the seller paid):

profit=selling price−cost priceprofit %=profitcost price×100%\begin{aligned} \text{profit} &= \text{selling price} - \text{cost price} \\[4pt] \text{profit \%} &= \frac{\text{profit}}{\text{cost price}} \times 100\% \end{aligned}

So:

  • a 20%20\% profit means selling price =1.2×= 1.2 \times cost price;
  • a 15%15\% loss means selling price =0.85×= 0.85 \times cost price.
cost priceprofitselling priceprofit % = profit ÷ cost price × 100
Profit on the cost priceThe percentage is always of the cost price

Marked price and discount

A discount is a reduction on the marked price, the price on the label. A shop may mark goods up and then give a discount. Its profit is still the selling price minus the cost price, as a percentage of the cost price.

marked priceselling pricedisc.discount % is of the marked price
DiscountTaken off the marked price

Worked example · WAEC 2021

WAEC 2021 · Paper 2 · Q7 (a)

The marked price for an item with a profit of 15%15\% is ₦60,000.00. If the item was sold at a discount of 5%5\%, calculate, correct to one decimal place, the percentage profit made on the transaction.

  1. The cost price

    • The marked price is 115%115\% of the cost price.
    • So the cost is 60 0001.15≈52 173.91{\frac{60\,000}{1.15} \approx 52\,173.91}, that is ₦52,173.91.

    Think first. The marked price includes a 15% profit. What percentage of the cost is it?

  2. The selling price

    • A 5%5\% discount leaves 95%95\% of the marked price.
    • 0.95×60 000=57 000{0.95 \times 60\,000 = 57\,000}, that is ₦57,000.

  3. The actual profit

    • Profit: 57 000−52 173.91=4826.09{57\,000 - 52\,173.91 = 4826.09}, that is ₦4,826.09.
    • As a percentage of the cost: 4826.0952 173.91×100%≈9.3%{\frac{4826.09}{52\,173.91} \times 100\% \approx 9.3\%}.

Two changes in a row

Each change applies to the amount after the change before it. So multiply the multipliers together.

10060× 0.642× 0.70.6 × 0.7 = 0.42: a 58% cut, not 70%
Two changes in a rowMultiply the multipliers: 0.6 × 0.7 = 0.42

Worked example · WAEC 2018

WAEC 2018 · Paper 2 · Q7

A shop had two reduction sales during which prices of all items were reduced by 40%40\% in the first sale and 30%30\% in the second. If a shirt was sold at GH₵ 35.00 during the second reduction sale, find the price before the first sale.

If the price of an article before the first reduction was GH₵ 180.00, find the total: (i) reduction in the price due to the two sales; (ii) percentage reduction in the price of the article.

  1. (a) The combined multiplier

    • A 40%40\% cut is ×0.6\times 0.6 and a 30%30\% cut is ×0.7\times 0.7: together 0.6×0.7=0.42{0.6 \times 0.7 = 0.42}.
    • So 0.42×original=35{0.42 \times \text{original} = 35}.
    • The original price was 350.42≈83.33{\frac{35}{0.42} \approx 83.33}, that is GH₵ 83.33.

    Think first. What do a 40% cut and then a 30% cut multiply the price by?

  2. (b)(i) The total reduction

    • After the first sale: 0.6×180=108{0.6 \times 180 = 108}.
    • After the second: 0.7×108=75.60{0.7 \times 108 = 75.60}.
    • Total reduction: 180−75.60=104.40{180 - 75.60 = 104.40}, that is GH₵ 104.40.

  3. (ii) As a percentage

    104.40180×100%=58%\frac{104.40}{180} \times 100\% = 58\%.

Your turn

WAEC 2019 · Paper 2 · Q10

  1. (a)

    A woman bought 130 kg of tomatoes for ₦52,000.00. She sold half of the tomatoes at a profit of 30%30\%. The rest of the tomatoes began to go bad; she then reduced the selling price per kg by 12%12\%. Calculate the new selling price per kg.

  2. (b)

    Calculate the percentage profit on the entire sales if she threw away 5 kg of bad tomatoes.

Worked solution (try it first)

(a)

  1. Cost per kg =52 000130=₦400= \frac{52\,000}{130} = ₦400.
  2. At 30%30\% profit the selling price was 1.3×400=₦5201.3 \times 400 = ₦520 per kg.
  3. Reduced by 12%12\%: 0.88×520=₦457.600.88 \times 520 = ₦457.60 per kg.

(b)

  1. She sold 65 kg at ₦520: ₦33,800.
  2. Of the other 65 kg she threw away 5 kg and sold 60 kg at ₦457.60: ₦27,456.
  3. Total sales =₦61,256= ₦61,256.
  4. Profit =61 256−52 000=₦9256= 61\,256 - 52\,000 = ₦9256.
  5. Percentage profit =925652 000×100%= \frac{9256}{52\,000} \times 100\%
    ≈17.8%\approx 17.8\%.

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