Discounts and Sale Prices
Master shop sales, find sale prices, work out savings, and reverse-engineer the original from a sale tag.
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A $\$180$ jacket is on sale for $\$126$. What percentage was taken off? Jot down your first reaction, then we'll see who's right.
Sales involve THREE quantities: marked (original) price, sale price, and discount. Knowing any two lets you find the third, both forwards and backwards.
A $\$180$ jacket is on sale for $\$126$. The discount in dollars is $180 - 126 = \$54$. The discount as a percentage is $\tfrac{54}{180} \times 100 = 30\%$. So this is "$30\%$ off". You can also reverse: knowing the sale price and the % off, find the original.
Know
- Marked price = original (RRP)
- Sale price = marked − discount
- Discount can be % or dollars
- Working backwards: marked = sale $\div$ (multiplier)
Understand
- Why discount % is always on the ORIGINAL (marked) price
- How to reverse to find the marked price from a sale price
- The difference between sale price and discount
Can Do
- Find sale price given % off
- Find % off given marked and sale prices
- Find marked price given sale price and discount %
Wrong: "Sale $\$126$, was $\$180$, so $\$126 \div \$180 = 70\%$ off", NO. $70\%$ is what you PAY, $30\%$ is OFF.
Right: You pay $70\%$ ($\$126$); discount is $30\%$ ($\$54$). Sum to $100\%$.
Wrong: "A jacket's sale is $\$60$ at $25\%$ off. So the marked price was $60 + 25 = \$85$.", NO; the $25\%$ is not $\$25$.
Right: Sale $= 75\%$ of marked, so marked $= 60 \div 0.75 = \$80$.
Forward problem, most common in shops. Given marked price and discount rate, find the sale price.
A $\$220$ guitar with $35\%$ off. Multiplier method: $\times (1 - 0.35) = \times 0.65$. $220 \times 0.65 = \$143$ sale price. The discount itself: $220 \times 0.35 = \$77$ saved. Check: $\$143 + \$77 = \$220$ ✓.
Given the sale price and discount rate, find the original (marked) price. Reverse the multiplier.
A jacket on sale for $\$84$, after $25\%$ off. The $\$84$ is $75\%$ of the marked price. So marked $= 84 \div 0.75 = \$112$. Saving: $\$112 - \$84 = \$28$. Always DIVIDE by the pay-fraction, never multiply.
Watch Me Solve It · 3 examples
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1Find discount in dollars$180 - 126 = \$54$Marked minus sale.
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2Discount as % of marked$\tfrac{54}{180} \times 100$Discount/marked × 100.
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3Compute$0.30 \times 100 = 30\%$So $30\%$ off.
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1Multiplier$1 - 0.40 = 0.60$You pay $60\%$.
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2Multiply$320 \times 0.60$Sale price.
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3Compute$320 \times 0.60 = \$192$Or: discount $= 0.40 \times 320 = 128$; $320-128=192$.
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1Sale = $70\%$ of marked$42 = 0.70 \times M$Pay-fraction is $1 - 0.30 = 0.70$.
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2Divide$M = 42 \div 0.70$Reverse: divide by pay-fraction.
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3Compute$M = \$60$Marked was $\$60$; you save $\$18$.
Common Pitfalls
Forward (Sale Price)
- Sale $= $ Marked $\times (1 - r)$
- $\$200$ at $25\%$ off $= \$150$
- Discount $= $ Marked $\times r$
Reverse (Marked Price)
- Marked $= $ Sale $\div (1 - r)$
- $\$150$ at $25\%$ off $\Rightarrow$ marked $= \$200$
- Divide, not multiply
Find % Off
- $\%$ off $= \tfrac{M - S}{M} \times 100$
- $\$80 \to \$60$ is $25\%$ off
- Always on the marked price
Quick Sanity Check
- Sale $+$ Saving $= $ Marked
- Pay $\%$ $+$ Off $\%$ $= 100\%$
- Sale $<$ Marked always
How are you completing this lesson?
Brain Trainer · 4 problems
Four drill problems to sharpen your skills. Work each, then reveal the answer.
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1 A $\$150$ shirt has $20\%$ off. Sale price?
$150 \times 0.80 = \$120$.$\$120$ -
2 Sale $\$36$, marked $\$60$. What % off?
$\tfrac{24}{60} \times 100 = 40\%$.$40\%$ off -
3 Sale $\$108$ after $10\%$ off. Marked?
$108 \div 0.90 = \$120$.$\$120$ -
4 A $\$45$ book is on sale for $\$36$. % off?
$\tfrac{9}{45} \times 100 = 20\%$.$20\%$ off
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. Calculate the sale price for: (a) $\$240$ marked, $35\%$ off; (b) $\$1450$ marked, $12\%$ off; (c) $\$76$ marked, $42\%$ off.
Q7. Tahlia paid $\$32$ for a top that was on sale for $20\%$ off. (a) What was the marked price? (b) How much did she save?
Q8. A guitar normally costs $\$480$. During a sale it is reduced by $25\%$. After the sale, the shop adds a further markup of $20\%$ on the sale price. (a) Find the post-sale price. (b) Is the new price equal to the original $\$480$? Show working. (c) Explain why a $25\%$ discount followed by a $20\%$ markup is NOT the same as ending up where you started.
Quick Check
1. B$\$120$.
2. B$20\%$.
3. C$\$180$.
4. B$30\%$.
5. C$\$90$.
Show Your Working Model Answers
Q6 (3 marks): (a) $240 \times 0.65 = \$156$ [1]. (b) $1450 \times 0.88 = \$1276$ [1]. (c) $76 \times 0.58 = \$44.08$ [1].
Q7 (2 marks): (a) Marked $= 32 \div 0.80 = \$40$ [1]. (b) Saved $= 40 - 32 = \$8$ [1].
Q8 (4 marks): (a) Sale: $480 \times 0.75 = \$360$. After markup: $360 \times 1.20 = \$432$ [2]. (b) No, $\$432 \neq \$480$ [1]. (c) The $25\%$ discount is on $\$480$ (savings $\$120$), but the $20\%$ markup is on the lower $\$360$ (only $\$72$). Different base prices mean the changes don't cancel, overall it's $\$48$ less, equivalent to a $10\%$ discount on the original [1].
The Layered Sale
A store advertises “Up to $50\%$ off!”. The fine print: $20\%$ off, then a further $25\%$ at the till, then a final $15\%$ student discount. (a) Find the overall multiplier and the equivalent single percentage discount. (b) Is the “$50\%$ off” advertising honest?
Reveal solution
(a) Combined multiplier $= 0.80 \times 0.75 \times 0.85 = 0.510$. So the equivalent single discount is $1 - 0.510 = 49\%$. (b) Very nearly $50\%$, just under. The ad is technically “up to”, which legally usually includes anything below, so it's arguable. The real saving is $49\%$, not the full $50\%$.
Three numbers
Marked, Sale, Discount
Forward
Sale = Marked × $(1-r)$
Reverse
Marked = Sale $\div (1-r)$
Find % off
Save/marked × 100
Sale $<$ Marked
Always
Sum check
Pay% + Off% = 100%
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