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Increases-And-Decreases Edexcel Higher

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Exam code:1MA1

Percentage increases & decreases

How do I increase by a percentage?

A percentage increase makes an amount bigger by adding that percentage on to itself
Without a calculator, use the basic percentages methods to find the percentage you are increasing by
- Then add this on to the original amount
- To increase 30 by 10%
  - 10% of 30 is 3
  - 30 + 3 = 33
  - This is equivalent to finding 110% of 30
With a calculator it is more efficient to use multipliers
- A multiplier is the decimal equivalent of a percentage
  - A percentage can be converted to a decimal by dividing by 100
- When increasing by a percentage, we are finding a percentage greater than 100%
- To increase 80 by 15%
  - We are finding 115% of 80, so the multiplier is 1.15
  - 1.15 × 80 = 92

How do I decrease by a percentage?

A percentage decrease makes an amount smaller by subtracting that percentage from itself
Without a calculator, use the methods outlined in Basic Percentages to find the percentage you are decreasing by
- Then subtract this from the original amount
- To decrease 30 by 10%
  - 10% of 30 is 3
  - 30 – 3 = 27
  - This is equivalent to finding 90% of 30
    - Because 100% – 10% = 90%
With a calculator it is more efficient to use multipliers
- When decreasing by a percentage, we are finding a percentage smaller than 100%
- To decrease 80 by 15%
  - We are finding 85% of 80, so the multiplier is 0.85
    - Because 100% – 15% = 85%
  - 0.85 × 80 = 68

Worked Example

(a) Increase 200 kg by 21%.

Method 1: Non-calculator

By first finding 10% and 1%, find 21% of 200

10% of 200 = 20
1% of 200 = 2
21% of 200 = 20 + 20 + 2 = 42

Add this to the original amount

200 + 42

242 kg

Method 2: Calculator

An increase by 21% is equivalent to finding 121% of the original amount
So the multiplier is 1.21

1.21 × 200

242 kg

(b) An item that costs £ 500 is discounted by 35%.

Find the new price of the item.

A discount of 35% means the price decreases by 35%

Method 1: Non-calculator

By first finding 10% and 5%, find 35% of 500

10% of 500 = 50
5% of 500 = 25
35% of 500 = 50 + 50 + 50 + 25 = 175

Subtract this from the original amount

500 – 175

£325

Method 2: Calculator

A decrease of 35% is equivalent to finding 65% of the original amount (100 – 35 = 65)
So the multiplier is 0.65

500 × 0.65

£ 325

How do I deal with repeated percentage changes?

In some problems there may be several changes by a percentage
For example,
- A shop increases the price of a product costing £80 by 10%,
  - equivalent to a multiplier of × 1.10
- and then discounts the product by 15%,
  - equivalent to a multiplier of × 0.85
- and then discounts the product by a further 20%
  - equivalent to a multiplier of × 0.80
You can either:
- Multiply the starting amount by each multiplier in turn
  - ( ( ( 80 × 1.10 ) × 0.85 ) × 0.80 ) = £59.84
- Or combine the multipliers first and then multiply by the “combined multiplier”
  - 1.10 × 0.85 × 0.80 = 0.748
    - This shows it is equivalent to 74.8% of the original amount, or a discount of 25.2%
  - 80 × 0.748 = £59.84
In general, for $n$ multipliers of values $m subscript 1 comma space m subscript 2 comma space... space comma space m subscript n$
- The combined multiplier is $m subscript 1 space cross times space m subscript 2 space cross times space... space cross times space m subscript n$

How do I find a percentage change?

The multiplier that was used for a percentage change can be found using the formula:
- <img alt=”m equals fraction numerator Amount space after over denominator Amount space before end fraction” data-mathml='<math ><semantics><mrow><mi>m</mi><mo>=</mo><mfrac><mrow><mi>Amount</mi><mo> </mo><mi>after</mi></mrow><mrow><mi>Amount</mi><mo> </mo><mi>before</mi></mrow></mfrac></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true}</annotation></semantics></math>’ height=”47″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2247%22%20width%3D%22150%22%20wrs%3Abaseline%3D%2230%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Em%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmi%3EAmount%3C%2Fmi%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmi%3Eafter%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3EAmount%3C%2Fmi%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmi%3Ebefore%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math17f39f8317fbdb1988ef4c628eb’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADRjdnQgDVUNBwAAAVAAAAA6Z2x5ZoPi2VsAAAGMAAAAsmhlYWQQC2qxAAACQAAAADZoaGVhCGsXSAAAAngAAAAkaG10eE2rRkcAAAKcAAAACGxvY2EAHTwYAAACpAAAAAxtYXhwBT0FPgAAArAAAAAgbmFtZaBxlY4AAALQAAABn3Bvc3QB9wD6AAAEcAAAACBwcmVwa1uragAABJAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACAAAAAEAAQAAQAAAD3%2F%2FwAAAD3%2F%2F%2F%2FEAAEAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr%2FasCAP4AA6v8VVUDAAACAIAA6wLVAhUAAwAHAGUYAbAIELAG1LAGELAF1LAIELAB1LABELAA1LAGELAHPLAFELAEPLABELACPLAAELADPACwCBCwBtSwBhCwB9SwBxCwAdSwARCwAtSwBhCwBTywBxCwBDywARCwADywAhCwAzwxMBMhNSEdASE1gAJV%2FasCVQHAVdVVVQAA

Maths Gcse Edexcel Higher