Worked example · Easy
Percentage to fraction
Write 35% as a fraction in simplest form.
\[35\%=\frac{35}{100}=\boxed{\frac7{20}}\]
Chapter 13
Percentages
Build fluent percentage skills from conversions and percentage-of-amount questions through reverse percentages, profit and loss, repeated change, and simple versus compound interest.
Fractions, decimals and percentagesPercentage of a quantityExpressing one quantity as a percentage of anotherPercentage changePercentage increase and decreasePercentage increase and decrease using a multiplierFinding the original quantityWages and salariesDiscount, profit and lossRepeated percentage changeCompound and simple interest
By the end of this chapter, you should be able to
- convert between fractions, decimals and percentages;
- find a percentage of a quantity and one quantity as a percentage of another;
- calculate percentage change, increases and decreases;
- use percentage multipliers efficiently;
- work backwards to find an original amount;
- solve wages, salaries, commission, discount, profit and loss problems;
- calculate repeated percentage change and compare simple with compound interest.
1
Subtopic 1
Fractions, decimals and percentages
Per cent means “out of 100”. A percentage can therefore be written as a fraction over 100, then simplified. To convert a percentage to a decimal, divide by 100.
Percentage÷ 100Decimal
Percentagewrite over 100Fraction
Worked example · Easy
Percentage to decimal
Write 2.7% as a decimal.
\[2.7\div100=\boxed{0.027}\]
Worked example · Medium
A percentage above 100%
Write 140% as a decimal and as a fraction in simplest form.
\[140\%=1.4\]\[140\%=\frac{140}{100}=\boxed{\frac75}\]
Remember for this subtopic
Use \(\frac{p}{100}\) for a percentage; convert using \(\text{decimal}=\frac{p}{100}\). An exact fraction should be simplified. For recurring decimals, first express the decimal as a fraction.
Worked example · Exam-style
Convert between all three forms
Write \(0.375\) as a fraction and a percentage.
Step-by-step solution
\(0.375=\frac{375}{1000}=\frac38\). Multiplying by 100 gives \(37.5\%\). Hence \(\boxed{\frac38=37.5\%}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Write \(0.42\) as a percentage.
Show Answer
Step-by-step solution
\(0.42\times100=\boxed{42\%}\).
2Medium practice [2 marks]
Write \(65\%\) as a fraction in simplest form.
Show Answer
Step-by-step solution
\(\frac{65}{100}=\frac{13}{20}\). Answer: \(\boxed{\frac{13}{20}}\).
3Medium practice [3 marks]
Write \(\frac7{16}\) as a decimal and percentage.
Show Answer
Step-by-step solution
\(7\div16=0.4375\), then \(0.4375\times100=\boxed{43.75\%}\).
4Exam-style practice [4 marks]
A student claims \(0.075=75\%\). Explain the error and correct it.
Show Answer
Step-by-step solution
Multiply the decimal by 100: \(0.075\times100=7.5\). The correct answer is \(\boxed{7.5\%}\), not 75%.
2
Subtopic 2
Percentage of a quantity
Without a calculator, useful percentages such as 10%, 5%, 20%, 25%, 50% and 75% can often be found mentally. With a calculator, multiplying by the decimal form is efficient.
Worked example · Easy
Use 10% and 5%
Find 15% of $240.
10% of 240 = 24 and 5% = 12.
\[24+12=\boxed{\$36}\]Worked example · Medium
Use a decimal multiplier
Find 37% of $48.
\[48\times0.37=\boxed{\$17.76}\]
Worked example · Exam-style
Small percentage of a large amount
Find 2.4% of $36 000.
\[36000\times0.024=\boxed{\$864}\]
Remember for this subtopic
To find \(p\%\) of a quantity, calculate \(\frac p{100}\times\text{quantity}\). Percentages above 100 are possible: they give a result larger than the starting value.
Worked example · Exam-style
A percentage of a quantity greater than 100%
A school orders \(125\%\) of its original 240 notebooks. How many notebooks does it order?
Step-by-step solution
\(125\%=1.25\); \(1.25\times240=\boxed{300\text{ notebooks}}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Find \(10\%\) of 340.
Show Answer
Step-by-step solution
\(340\div10=\boxed{34}\).
2Medium practice [2 marks]
Find \(17\%\) of 250.
Show Answer
Step-by-step solution
\(\frac{17}{100}\times250=\boxed{42.5}\).
3Medium practice [3 marks]
Find \(2.5\%\) of 720.
Show Answer
Step-by-step solution
\(0.025\times720=\boxed{18}\).
4Exam-style practice [4 marks]
A company produces 480 items. It rejects \(7.5\%\). How many items are rejected?
Show Answer
Step-by-step solution
\(7.5\%=0.075\). Then \(480\times0.075=\boxed{36}\). There are 36 rejected items.
3
Subtopic 3
Expressing one quantity as a percentage of another
Formula
\[\text{percentage}=\frac{\text{first quantity}}{\text{second quantity}}\times100\%\]If units differ, convert them first.
Worked example · Easy
Simple comparison
Express 4 as a percentage of 5.
\[\frac45\times100=\boxed{80\%}\]
Worked example · Medium
Different units
Express 70 cm as a percentage of 2.3 m.
2.3 m = 230 cm.
\[\frac{70}{230}\times100=30.434\ldots\%\]\(\boxed{30.4\%}\) to 1 d.p.
Remember for this subtopic
Use the same units before dividing. The denominator is the quantity named after “of” or “out of”: \(\text{part}\div\text{whole}\times100\%\).
Worked example · Exam-style
Compare quantities with different units
How much of a 2.5 kg bag is 650 g, expressed as a percentage?
Step-by-step solution
Convert \(2.5\text{ kg}=2500\text{ g}\). \(\frac{650}{2500}\times100=\boxed{26\%}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
9 out of 30 students are absent. What percentage is absent?
Show Answer
Step-by-step solution
\(\frac9{30}\times100=\boxed{30\%}\).
2Medium practice [2 marks]
Express 18 as a percentage of 72.
Show Answer
Step-by-step solution
\(\frac{18}{72}\times100=\boxed{25\%}\).
3Medium practice [3 marks]
Express 45 cm as a percentage of 1.5 m.
Show Answer
Step-by-step solution
First convert 1.5 m to 150 cm. \(\frac{45}{150}\times100=\boxed{30\%}\).
4Exam-style practice [4 marks]
The price of a notebook is $18 and a pen is $12. Express the notebook price as a percentage of the pen price.
Show Answer
Step-by-step solution
Use the pen price as the base: \(\frac{18}{12}\times100=\boxed{150\%}\).
4
Subtopic 4
Percentage change
Original
100%
100%
→ change →
New amount
Use the original amount underneath the fraction.Percentage change
\[\text{percentage change}=\frac{\text{change}}{\text{original amount}}\times100\%\]Worked example · Medium
Percentage profit
A painting is bought for $45 and sold for $72. Find the percentage profit.
\[\text{profit}=72-45=27\]\[\frac{27}{45}\times100=\boxed{60\%}\]
Worked example · Exam-style
Percentage decrease
A computer falls in value from $1200 to $700. Find the percentage decrease.
\[\text{decrease}=1200-700=500\]\[\frac{500}{1200}\times100=41.666\ldots\%\]
\(\boxed{41.7\%}\) to 1 d.p.
Remember for this subtopic
Percentage change uses the ORIGINAL quantity as the denominator: \(\frac{\text{new}-\text{old}}{\text{old}}\times100\). State whether the result is an increase or decrease.
Worked example · Exam-style
Percentage change after a fall
Attendance falls from 480 to 414. Calculate the percentage decrease.
Step-by-step solution
Fall \(=480-414=66\). Percentage decrease \(=\frac{66}{480}\times100=\boxed{13.75\%}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
A price rises from $40 to $50. Find the percentage increase.
Show Answer
Step-by-step solution
Increase \(=50-40=10\). \(\frac{10}{40}\times100=\boxed{25\%}\).
2Medium practice [2 marks]
A value falls from 200 to 170. Find the percentage decrease.
Show Answer
Step-by-step solution
Decrease \(=30\). \(\frac{30}{200}\times100=\boxed{15\%}\).
3Medium practice [3 marks]
Exam marks improve from 56 to 70. Find the percentage increase.
Show Answer
Step-by-step solution
Increase \(=14\). \(\frac{14}{56}\times100=\boxed{25\%}\).
4Exam-style practice [4 marks]
A club membership increases from 125 to 150 and then decreases to 135. Find the overall percentage change from the start.
Show Answer
Step-by-step solution
Change from 125 to 135 is \(+10\). \(\frac{10}{125}\times100=\boxed{8\%\text{ increase}}\). Use the original 125, not 150.
5
Subtopic 5
Percentage increase and decrease
One method is to calculate the increase or decrease separately, then add or subtract it from the original amount.
Worked example · Easy
Increase a salary
A salary of $24 000 increases by 3%.
\[24000\times0.03=720\]\[24000+720=\boxed{\$24720}\]
Worked example · Medium
Sale reduction
An item costs $17.60 and is reduced by 15%.
\[17.60\times0.15=2.64\]\[17.60-2.64=\boxed{\$14.96}\]
Remember for this subtopic
To increase a number by \(p\%\), add \(\frac p{100}\) of it. To decrease it, subtract that percentage of the ORIGINAL amount.
Worked example · Exam-style
Sale-price and subsequent increase
A jacket costing $80 is reduced by 15%. Calculate the sale price.
Step-by-step solution
Discount \(=0.15\times80=12\). New price \(=80-12=\boxed{\$68}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Increase 60 by 20%.
Show Answer
Step-by-step solution
Increase \(=0.20\times60=12\). New value \(=\boxed{72}\).
2Medium practice [2 marks]
Decrease $90 by 30%.
Show Answer
Step-by-step solution
Reduction \(=0.30\times90=27\). New value \(=\boxed{\$63}\).
3Medium practice [3 marks]
A population of 2400 increases by 12%. Find the new population.
Show Answer
Step-by-step solution
Increase \(=0.12\times2400=288\). New population \(=\boxed{2688}\).
4Exam-style practice [4 marks]
A printer costs $250. It is first reduced by 12% and then a delivery fee of $15 is added. Find the total.
Show Answer
Step-by-step solution
Discount \(=0.12\times250=30\). Sale price \(250-30=220\). Add $15: \(\boxed{\$235}\).
6
Subtopic 6
Percentage increase and decrease using a multiplier
Increase by 8%100% + 8% = 108%× 1.08
Decrease by 15%100% − 15% = 85%× 0.85
Worked example · Easy
One-step increase
A salary of $17 000 increases by 3%.
Multiplier = 1.03.
\[17000\times1.03=\boxed{\$17510}\]Worked example · Medium
One-step decrease
A $155 tablet is reduced by 15%.
Multiplier = 0.85.
\[155\times0.85=\boxed{\$131.75}\]Fast multiplier rule
Increase by \(p\%\): multiply by \(1+\frac p{100}\). Decrease by \(p\%\): multiply by \(1-\frac p{100}\).
Remember for this subtopic
A percentage multiplier makes the calculation one step: \(1+\frac p{100}\) for an increase; \(1-\frac p{100}\) for a decrease. The multiplier multiplies the old amount.
Worked example · Exam-style
Use multipliers to find a sale price
A bicycle costs $375 and is discounted by 16%. Find its new price using a multiplier.
Step-by-step solution
Multiplier \(=1-0.16=0.84\). New price \(375\times0.84=\boxed{\$315}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Write the multiplier for a 7% increase.
Show Answer
Step-by-step solution
\(1+\frac7{100}=\boxed{1.07}\).
2Medium practice [2 marks]
Write the multiplier for a 35% decrease.
Show Answer
Step-by-step solution
\(1-\frac{35}{100}=\boxed{0.65}\).
3Medium practice [3 marks]
Increase 550 by 18% in one step.
Show Answer
Step-by-step solution
\(550\times1.18=\boxed{649}\).
4Exam-style practice [4 marks]
After a 22% reduction a machine costs $624. Find its cost BEFORE the reduction.
Show Answer
Step-by-step solution
Final multiplier is \(0.78\). Original price \(=624\div0.78=\boxed{\$800}\).
7
Subtopic 7
Finding the original quantity
Reverse percentages work backwards. If the final amount was found by multiplying, recover the original by dividing by the multiplier.
Original
× 1.20→÷ 1.20
Final
Worked example · Medium
Reverse an increase
After a 20% pay rise, a salary is $31 260. Find the original salary.
\[\text{original}=31260\div1.20=\boxed{\$26050}\]
Worked example · Exam-style
Reverse a sale discount
After a 10% reduction, a dress costs $38.70. Find its original price.
The sale price is 90% of the original, so multiplier = 0.90.
\[38.70\div0.90=\boxed{\$43}\]Common mistake
If a sale price is 20% less than the original, do not add 20% to the sale price. The sale price represents 80% of the original, so divide by 0.80.
Remember for this subtopic
For reverse percentages, divide the FINAL amount by the multiplier. A final price after a 20% reduction represents 80% of the original, not 120%.
Worked example · Exam-style
Find an original value after a rise
After an 18% increase, a salary is $2950. Find the original salary.
Step-by-step solution
\(\text{original}\times1.18=2950\). \(\text{original}=2950\div1.18=\boxed{\$2500}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
A price including 20% tax is $120. Find the price before tax.
Show Answer
Step-by-step solution
\(\text{original}=120\div1.20=\boxed{\$100}\).
2Medium practice [2 marks]
After a 10% discount a bag costs $45. Find the original price.
Show Answer
Step-by-step solution
\(45\div0.90=\boxed{\$50}\).
3Medium practice [3 marks]
A population after increasing by 25% is 1000. Find the starting population.
Show Answer
Step-by-step solution
\(1000\div1.25=\boxed{800}\).
4Exam-style practice [4 marks]
A shop sells a watch for $161 after a 30% reduction. Calculate its original price.
Show Answer
Step-by-step solution
Final amount is 70%. Original price \(=161\div0.7=\boxed{\$230}\).
8
Subtopic 8
Wages and salaries
Wage
Usually depends on hours or days worked.
Salary
A fixed amount, often quoted per year and paid monthly.
Gross pay is pay before deductions. Take-home pay is what remains after tax and other deductions. Some jobs include commission, usually a percentage of sales.
Worked example · Easy
Monthly salary
An annual salary is $8500. Find the monthly gross pay.
\[8500\div12=708.333\ldots\]
\(\boxed{\$708.33}\) to the nearest cent.
Worked example · Medium
Basic pay plus commission
Basic annual pay is $9600. A worker also earns 5% commission on monthly sales. In May, sales are $7500. Find May gross pay.
\[9600\div12=800\]\[0.05\times7500=375\]\[800+375=\boxed{\$1175}\]
Remember for this subtopic
Total pay may combine a basic wage, overtime and a commission. Be careful whether commission is a percentage of SALES or a percentage of PAY.
Worked example · Exam-style
Basic pay plus overtime and commission
A worker earns $12 per hour for 35 hours, overtime at $18 per hour for 5 hours, and $30 commission. Find total earnings.
Step-by-step solution
Basic \(35\times12=420\). Overtime \(5\times18=90\). Total \(420+90+30=\boxed{\$540}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
A worker earns $11 per hour for 8 hours. Calculate earnings.
Show Answer
Step-by-step solution
\(11\times8=\boxed{\$88}\).
2Medium practice [2 marks]
A salesperson receives 4% commission on $3500 sales. Find the commission.
Show Answer
Step-by-step solution
\(0.04\times3500=\boxed{\$140}\).
3Medium practice [3 marks]
A person works 30 hours at $15 per hour and 4 overtime hours at $22 per hour. Find total pay.
Show Answer
Step-by-step solution
Basic \(30\times15=450\). Overtime \(4\times22=88\). Total \(=\boxed{\$538}\).
4Exam-style practice [4 marks]
A salesperson gets a $900 basic wage plus 3% of sales. Total pay is $1110. Find the sales.
Show Answer
Step-by-step solution
Commission \(1110-900=210\). Sales \(=210\div0.03=\boxed{\$7000}\).
9
Subtopic 9
Discount, profit and loss
Discount
Reduction from the original price.
Profit
Selling price − cost price.
Loss
Cost price − selling price.
Important denominator
\[\text{percentage profit or loss}=\frac{\text{profit or loss}}{\text{cost price}}\times100\%\]Worked example · Easy
Discount
A jacket costs $38 and has a 15% discount.
The customer pays 85%.
\[38\times0.85=\boxed{\$32.30}\]Worked example · Medium
Percentage profit
An item costs $1.40 and sells for $1.80. Find the percentage profit.
\[\text{profit}=0.40\]\[\frac{0.40}{1.40}\times100=28.571\ldots\%\]
\(\boxed{28.6\%}\) to 1 d.p.
Worked example · Exam-style
Percentage loss
A car bought for $9500 is sold for $8300. Find the percentage loss.
\[\text{loss}=1200\]\[\frac{1200}{9500}\times100=12.631\ldots\%\]
\(\boxed{12.6\%}\) to 1 d.p.
Remember for this subtopic
Percentage profit or loss is calculated relative to COST price, while a discount is normally a percentage of the ORIGINAL marked selling price. Keep those bases separate.
Worked example · Exam-style
Profit percentage after a discount
A shop buys a bag for $48 and marks it at $75. It offers a 20% discount. Find the percentage profit.
Step-by-step solution
Sale price \(=75\times0.80=60\). Profit \(=60-48=12\). Percentage profit \(=\frac{12}{48}\times100=\boxed{25\%}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
An item costs $80 and sells for $100. Find the percentage profit.
Show Answer
Step-by-step solution
Profit \(=20\). \(\frac{20}{80}\times100=\boxed{25\%}\).
2Medium practice [2 marks]
An item bought for $150 sells for $120. Find the percentage loss.
Show Answer
Step-by-step solution
Loss \(=30\). \(\frac{30}{150}\times100=\boxed{20\%}\).
3Medium practice [3 marks]
A jacket is marked $200 and discounted by 35%. Find its sale price.
Show Answer
Step-by-step solution
\(200\times0.65=\boxed{\$130}\).
4Exam-style practice [4 marks]
A trader buys an item for $360 and wants a 15% profit on cost. Find the selling price.
Show Answer
Step-by-step solution
\(360\times1.15=\boxed{\$414}\).
10
Subtopic 10
Repeated percentage change
For repeated increases or decreases, use the same multiplier again for every period. This is why powers are useful.
Start× 1.05Year 1× 1.05Year 2× 1.05Year 3
Worked example · Medium
Repeated increase
An item costs $12 and rises by 5% each year. Find the price after 2 years.
\[12\times1.05^2=\boxed{\$13.23}\]
Worked example · Exam-style
Repeated depreciation
A machine worth $3500 loses 11% of its value each year. Find its value after 4 years.
Multiplier = \(1-0.11=0.89\).
\[3500\times0.89^4=2195.968\ldots\]\(\boxed{\$2195.97}\) to the nearest cent.
Remember for this subtopic
Repeated percentage changes multiply: \(A=P(1\pm\frac p{100})^n\). A rise by 10% followed by a fall by 10% does NOT return to the original amount.
Worked example · Exam-style
Different successive rates
A device costs $800. Its price increases by 15%, then decreases by 10%. Find the final price.
Step-by-step solution
\(800\times1.15\times0.9=\boxed{\$828}\). Overall change \(=\frac{828-800}{800}\times100=3.5\%\text{ increase}\).
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Increase 100 by 10% twice.
Show Answer
Step-by-step solution
\(100\times1.1^2=\boxed{121}\).
2Medium practice [2 marks]
Decrease 600 by 20% twice.
Show Answer
Step-by-step solution
\(600\times0.8^2=\boxed{384}\).
3Medium practice [3 marks]
A machine worth $4000 loses 15% in value each year. Find its value after 2 years.
Show Answer
Step-by-step solution
\(4000\times0.85^2=\boxed{\$2890}\).
4Exam-style practice [4 marks]
A population rises by 5% for 3 years from 8000. Find the population after 3 years, nearest whole number.
Show Answer
Step-by-step solution
\(8000\times1.05^3=9261\). Answer \(\boxed{9261}\) people.
11
Subtopic 11
Compound and simple interest
Simple interest
Interest is calculated on the original principal every time.
4000420044004600
Compound interest
Interest is added first, then future interest is calculated on the new balance.
4000420044104630.50
Compound-interest formula
\[A=P\left(1+\frac r{100}\right)^n\]where \(P\) is the principal, \(r\) is the percentage rate per period, \(n\) is the number of periods and \(A\) is the final amount.
Worked example · Easy
Simple interest
$4000 is invested for 2 years at 5% simple interest per year.
Interest per year:
\[4000\times0.05=200\]Two years' interest:
\[2\times200=400\]Final amount:
\[4000+400=\boxed{\$4400}\]Worked example · Medium
Compound interest
$4000 is invested for 6 years at 5% compound interest.
\[4000(1.05)^6=5360.3825\ldots\]
\(\boxed{\$5360.38}\)
Worked example · Exam-style
Compare two investments
$4000 is invested for 5 years. Option A pays 3.5% simple interest; Option B pays 3.25% compound interest. Which is better?
Option A:
\[4000+(4000\times0.035\times5)=4700\]Option B:
\[4000(1.0325)^5=4693.65\text{ (to nearest cent)}\]Option A is better by
\[4700-4693.65=\boxed{\$6.35}\]Remember for this subtopic
Simple interest is calculated on the original principal each year: \(I=Prt/100\). Compound interest is calculated on the increasing balance: \(A=P(1+r/100)^t\).
Worked example · Exam-style
Simple versus compound interest
$2000 is invested at 5% per year for three years. Compare the simple- and compound-interest totals.
Step-by-step solution
Simple: \(2000(1+0.05\times3)=\boxed{\$2300}\). Compound: \(2000(1.05)^3=\boxed{\$2315.25}\). Compound gives $15.25 more.
Progressive practice: confidence to exam-style
Try each question before opening the solution. Exam-style questions are original and modelled on the skills tested in examinations.
1Easy practice [1 marks]
Find simple interest on $1000 at 4% per year for 2 years.
Show Answer
Step-by-step solution
\(I=1000\times0.04\times2=\boxed{\$80}\).
2Medium practice [2 marks]
Find the amount after 2 years for $500 at 10% compound interest per year.
Show Answer
Step-by-step solution
\(500\times1.1^2=\boxed{\$605}\).
3Medium practice [3 marks]
A loan of $600 at simple interest earns $90 over 3 years. Find the annual rate.
Show Answer
Step-by-step solution
\(90=600\times r\times3/100\). \(r=90\times100/(1800)=\boxed{5\%}\).
4Exam-style practice [4 marks]
$800 is deposited at 2.5% compound interest per year for 2 years. Find the interest earned, to the nearest cent.
Show Answer
Step-by-step solution
Final amount \(=800(1.025)^2=840.50\). Interest \(=840.50-800=\boxed{\$40.50}\).
Chapter 13 practice
Questions begin with percentage fluency and progress to reverse percentages, profit/loss and repeated change. Try each question before opening the solution.
A. Core percentage skills
1Convert a percentage[2 marks]
Write 62.5% as a decimal and a fraction in simplest form.
Show Answer
\[62.5\%=0.625\]\[0.625=\frac{625}{1000}=\boxed{\frac58}\]
2Percentage of an amount[3 marks]
Find 32% of $720.
Show Answer
\[720\times0.32=\boxed{\$230.40}\]
3One quantity as a percentage[3 marks]
Express $1.70 as a percentage of $2.
Show Answer
\[\frac{1.70}{2}\times100=\boxed{85\%}\]
4Units first[4 marks]
Express 45 cm as a percentage of 1.8 m.
Show Answer
1.8 m = 180 cm.
\[\frac{45}{180}\times100=\boxed{25\%}\]B. Percentage change and multipliers
5Percentage increase[4 marks]
A wage rises from $5.00 to $5.60 per hour. Find the percentage increase.
Show Answer
\[\text{increase}=0.60\]\[\frac{0.60}{5.00}\times100=\boxed{12\%}\]
6Percentage decrease[4 marks]
A value falls from $1250 to $600. Find the percentage decrease.
Show Answer
\[\text{decrease}=650\]\[\frac{650}{1250}\times100=\boxed{52\%}\]
7Increase using a multiplier[3 marks]
Increase $2500 by 6.2%.
Show Answer
Multiplier = 1.062.
\[2500\times1.062=\boxed{\$2655}\]8Decrease using a multiplier[3 marks]
A $42 pair of shoes is reduced by 30%. Find the sale price.
Show Answer
Multiplier = 0.70.
\[42\times0.70=\boxed{\$29.40}\]C. Reverse percentages and money
9Reverse an increase[4 marks]
After a 12% increase, a quantity is 84 tonnes. Find the original quantity.
Show Answer
\[\text{original}=84\div1.12=\boxed{75\text{ tonnes}}\]
10Reverse a decrease[4 marks]
After a 5% reduction, a pair of shoes costs $47.50. Find the original price.
Show Answer
47.50 is 95% of the original.
\[47.50\div0.95=\boxed{\$50}\]11Commission[4 marks]
A worker earns an annual basic salary of $12 000 plus 7.5% commission on annual sales of $76 000. Find total annual pay.
Show Answer
\[0.075\times76000=5700\]\[12000+5700=\boxed{\$17700}\]
12Percentage profit[4 marks]
An article is bought for $22 and sold for $27. Find the percentage profit.
Show Answer
\[\text{profit}=5\]\[\frac5{22}\times100=22.727\ldots\%\]
\(\boxed{22.7\%}\) to 1 d.p.
D. Exam-style challenge
13Repeated increase[5 marks]
Shares worth $8000 increase by 7.5% each year. Find their value after 4 years.
Show Answer
\[8000(1.075)^4=10684.04\ldots\]
\(\boxed{\$10684.04}\)
14Repeated depreciation[5 marks]
A car worth $10 500 falls in value by 12% per year. Find its value after 3 years.
Show Answer
Multiplier = 0.88.
\[10500(0.88)^3=7158.0672\]\(\boxed{\$7158.07}\)
15Compound interest[5 marks]
$1500 is invested at 4% compound interest for 4 years. Find the final amount.
Show Answer
\[1500(1.04)^4=1754.78784\]
\(\boxed{\$1754.79}\)
16Simple versus compound[7 marks]
$500 is invested for 3 years. Compare 12% simple interest with 10% compound interest. Which earns more interest, and by how much?
Show Answer
Simple interest:
\[500\times0.12\times3=180\]Compound final amount:
\[500(1.10)^3=665.50\]Compound interest:
\[665.50-500=165.50\]Simple interest earns more by
\[180-165.50=\boxed{\$14.50}\]17Reverse tax[6 marks]
A television costs $564 including 17.5% sales tax. Find the price before tax.
Show Answer
The taxed price is 117.5% of the pre-tax price.
\[564\div1.175=\boxed{\$480}\]18Mixed repeated rates[7 marks]
An investment of $3500 earns 5% compound interest for 3 years and then 7.5% for the next 3 years. Find its value after 6 years.
Show Answer
\[3500(1.05)^3(1.075)^3=5016.03\ldots\]
\(\boxed{\$5016.03}\) to the nearest cent.
Chapter 13 high-grade checklist
Conversions
I can move between fractions, decimals and percentages without confusing ×100 and ÷100.
Original amount
I use the original amount as the denominator for percentage change, profit and loss.
Multipliers
I can turn any percentage increase/decrease into a one-step multiplier.
Reverse percentages
I divide by the final multiplier to recover an original amount.
Money contexts
I can handle salary, commission, discount, profit and loss with the correct base amount.
Repeated change
I use powers for repeated percentage change and understand why compound interest grows differently from simple interest.