Solving Problems with Percentages

Students learn how to convert between fractions, decimals and percentages and how to write one number as a percentage of another.  They use this knowledge to calculate a repeated percentage  change and reverse percentages.

This unit takes place in Term 1 of Year 10 and follows on from working with fractions and decimals.

Prerequisite Knowledge
• Multiply and divide by powers of ten
• Recognise the per cent symbol (%)
• Understand that per cent relates to ‘number of parts per hundred’
• Write one number as a fraction of another
• Calculate equivalent fractions

Success Criteria
• Define percentage as ‘number of parts per hundred
• Interpret fractions and percentages as operators
• Interpret percentages as a fraction or a decimal
• Interpret percentages changes as a fraction or a decimal
• Interpret percentage changes multiplicatively
• Express one quantity as a percentage of another
• Compare two quantities using percentages
• Work with percentages greater than 100%;
• Solve problems involving percentage change
• Solve problems involving percentage increase/decrease
• Solve problems involving original value problems
• Solve problems involving simple interest including in financial mathematics
• Set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes.

Key Concepts
• Use the place value table to illustrate the equivalence between fractions, decimals and percentages.
• To calculate a percentage of an amount without calculator students need to be able to calculate 10% of any number by dividing by 10.
• To calculate a percentage of an amount with a calculator students should be able to convert percentages to decimals mentally and use the percentage function.
• Equivalent ratios are useful for calculating the original amount after a percentage change.
• To calculate the multiplier for a percentage change students need to understand 100% as the original amount. E.g., 10% decrease represents 10% less than 100% = 0.9.
• Students need to have a secure understanding of the difference between simple and compound interest.

Common Misconceptions
• Students often consider percentages to limited to 100%. A key learning point is to understand how percentages can exceed 100%.
• Students sometimes confuse 70% with a magnitude of 70 rather than 0.7.
• Students can confuse 65% with 1/65 rather than 65/100.
• Compound interest is often confused with simple interest, i.e., 10% compound interest = 110% × 110% = 1.12 not 220% (2.2).

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