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Adding and Taking off a Percentage: Level 9

Indicator of progress

A very common use of percentages is to increase or decrease a given amount by a percentage. Many students will do this by calculating the mark-up or discount separately, and then adding or subtracting from the price.

Success at this level depends on students being able to add or subtract a percentage in one step by multiplication.

For example, to find a selling price after a 17% mark-up has been added, multiply by (1 + 0.17); to find a sale price when a 30% discount is given, multiply by (1 - 0.30). For more information,

For more information, see: More about adding Percentages - (Level 9)

 

Illustration 1: Adding percentages by multiplication by a decimal

At this stage, students will already be able to calculate the result by calculating the mark up first. For example, to find the price after a 17% mark-up has been added to a wholesale price of $80, they will find the mark up ($80 × 0.17 = $13.60) and then add to get the final price ($80 + $13.60). They will not know they can multiply the original price by (1 + 0.17).

To find their pay if 27% is subtracted for tax, they will calculate the tax and subtract from the pay. They will not know they can multiply the pay by (1 - 0.27).

 

Teaching strategies

The first two teaching strategies focus on developing the meaning and equivalence of adding percentages and multiplying by the appropriate decimal. The remaining activities provide consolidation and practice.

Activity 1: Check pre-requisite knowledge illustrates that this is a high level concept.

Activity 2: A mental model for percentages over 100% demonstrates and establishes the equivalences using lengths.

Activity 3: Matching operations consolidates the equivalence.

Activity 4: Target 100 and Activity 5 : Using calculators and spreadsheets focus on transferring these ideas to technology.

Activity 6: An intriguing problem where the answers are the same both ways, requires equivalence as its explanation; adding percentages is just multiplying, so the order doesn't matter.

 

Activity 1: Check pre-requisite knowledge

Success with this new approach depends on a range of pre-requisite knowledge. Check the following are in place.

1. Converting percentages <100% to decimals. To gauge the understanding of the students ask them to convert percentages like 12%, 12.3%, 5%, 12%, 7.5% to decimals. Ask them to explain how they did it and why it works.

2. Knowing about percentages over 100%. For example, ask students to calculate mentally 150% of an even number eg $12. Again ask about how they worked it out and why it works. Ask students when they might see percentages over 100% used in everyday life?

3. Multiplying whole numbers by decimals. For example can students estimate or calculate the answer to missing number problems like 100 × ? = 120; 200 × ? = 175 ; again ask about how and why.

4. Calculating a percentage of a given amount? For example ask students to estimate what would be: 5% of 250; 15% of 3500 ; again ask about how and why.

 

Activity 2: A mental model for percentages over 100%

  • Draw a line. Measure its length in mm. This line is 100%.
  • Work out 10% of the line length. Draw that much extra at the end of the line.
  • Measure the new total length.
  • Now work out 110% of the original length, and compare with the measurement.
  • Now work out the 1.1 × original length. Compare and think about why.
  • They should compare their work with others who should have the same general answer, but used a different line length – that is, it always works.

Repeat for removal of 10% to give 90% (and 0.9) × original length

 

Activity 3: Matching operations

Students should match each entry in the left hand column with an entry in the right hand column. For example, to find 43% of an amount, you multiply the amount by 0.43 . Students should test answers they are unsure about with a special case. For example, if they want to know if multiplying by 0.95 is the same as subtracting 5%, they can test it on $100. The group should be encouraged to discuss why this works following completion of the task.

This table can be supplemented by suggestions from students. Click on the table title to access a PDF document that can be downloaded and printed off for students.

Match up the operations that have the same effect (PDF - 15Kb)

find 43%

multiply by 0.95

add 10%

multiply by 0.97

add 3%

multiply by 0.43

find 10%

multiply by 1.13

subtract 5%

multiply by 2.1

add 100%

multiply by 1.5

add half the amount

multiply by 0.03

subtract 3%

multiply by 1.01

find 3%

multiply by 1.1

add 143%

double

subtract 43%

multiply by 0.57

add 43%

multiply by 0.1

add 110%

multiply by 1.43

add 13%

multiply by 1.03

add 1%

multiply by 2.43

 

Answers

 

Match up the operations that have the same effect

find 43%

multiply by 0.95

add 10%

multiply by 0.97

add 3%

multiply by 0.43

find 10%

multiply by 1.13

subtract 5%

multiply by 2.1

add 100%

multiply by 1.5

add half the amount

multiply by 0.03

subtract 3%

multiply by 1.01

find 3%

multiply by 1.1

add 143%

double

subtract 43%

multiply by 0.57

add 43%

multiply by 0.1

add 110%

multiply by 1.43

add 13%

multiply by 1.03

add 1%

multiply by 2.43

 

Activity 4: Calculator Game: Target 100

calculator

The intention of this calculator game is that students get a sense of what numbers such as 95 (or 105) must be multiplied by to get close to 100, that is, familiarity with multiplying by numbers close to 1.

Using one calculator, one player enters a number in a given range (say between 15 and 20), then the players take turns to multiply the number on the screen to get the answer close to 100. The answer must be between 100 and 101. Even when the number is above 100, the players must only multiply.

 

Activity 5: Using calculators and spreadsheets

The purpose of this activity is for the students to learn how to use their calculators for common percentage problems. Since calculators differ, they will need to experiment and check.

Ask students to use a calculator with a percentage key, to work out how to calculate the answer to a problem like this:

“My football team had 2000 members last year. There is a 15% increase in membership this year. How many members are there now?”

Note: On my calculator, pressing the buttons "2000 + 15% = " in sequence adds 15% to 2000 and pressing the buttons "2000 - 15% =" in sequence subtracts 15% from 2000. Ask students to compare the calculator methods with the decimal method (2000 × 1.15). Pose other problems and ask them to calculate the answer both ways.

 

Activity 6: An intriguing problem

At a shop, I can get a 15% discount with my special card, but I have to pay an import tax of 8%.

Should I ask them to give me the discount first and then charge the import tax on the lower discounted price, or should I ask them to first charge me the import tax and then give me a bigger discount?

Does the best strategy for me depend on how much I am spending, or on the exact rates of discount and tax?

The key to solving this problem is appreciating that adding the tax and taking off the discount are both just multiplications. Ask students to guess which is best, and to explain their reasons to others before any calculations are done.