Fractions, Decimals and PercentagesAQA GCSE Maths: Flashcards

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What is a terminating decimal and how does it relate to fractions?

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What is a terminating decimal and how does it relate to fractions?: A terminating decimal is a decimal that ends after a finite number of digits. It can be expressed as a fraction with a denominator that has only factors of 2 and 5. For example, 0.25 = 1/4.
Define a recurring decimal and give an example.: A recurring decimal is a decimal in which one or more digits repeat infinitely. It is denoted with a dot or bar over the repeating digits. Example: 0.333... = 1/3 or 0.142857142857... = 1/7.
How do you convert a recurring decimal to a fraction? (Higher Tier): Let x equal the recurring decimal. Multiply both sides by an appropriate power of 10 to align the recurring parts, then subtract the original equation from the new equation to eliminate the recurring part. Solve for x as a fraction. Example: If x = 0.333..., then 10x = 3.333..., so 10x - x = 3, giving 9x = 3, thus x = 1/3.
How do you convert a fraction to a recurring decimal?: Divide the numerator by the denominator using long division. If the division never terminates and digits begin to repeat, the result is a recurring decimal. The remainder pattern determines which digits repeat.
What is the definition of a percentage?: A percentage is a number expressed as a fraction of 100. The symbol % means 'per hundred'. For example, 25% = 25/100 = 1/4.
Explain what it means to interpret a fraction as an operator.: Interpreting a fraction as an operator means using it to multiply a quantity. For example, 3/5 of 200 means (3/5) × 200 = 120. The fraction acts as a multiplier.
Explain what it means to interpret a percentage as an operator.: Interpreting a percentage as an operator means using it to find a proportion of a quantity. For example, 15% of 80 means 0.15 × 80 = 12. The percentage is converted to a decimal multiplier.
How do you identify and work with fractions in ratio problems?: In a ratio problem, fractions represent parts of a whole. For example, if a ratio is 3:2, the fractions are 3/5 and 2/5 of the total. Use fractions to find actual quantities when given the total or individual parts.
What is the formula for percentage change?: Percentage change = (Actual change / Original value) × 100%. Example: If a price goes from £50 to £60, the change is £10, so percentage change = (10/50) × 100% = 20%.
Explain the difference between percentage increase and percentage decrease.: Percentage increase occurs when a value grows: New value = Original × (1 + percentage/100). Percentage decrease occurs when a value falls: New value = Original × (1 - percentage/100). Example: A 10% increase on £100 = £110; a 10% decrease on £100 = £90.
How do you solve an original value problem (reverse percentage)?: If you know the new value after a percentage change, divide by the percentage multiplier to find the original value. Example: If a value after a 20% increase is £120, then original value = £120 ÷ 1.2 = £100. Use multiplier (1 + percentage/100) for increases or (1 - percentage/100) for decreases.
What is simple interest and what is its formula?: Simple interest is interest calculated only on the original principal amount, not on accumulated interest. Formula: Interest = (Principal × Rate × Time) / 100, or I = PRT/100. The rate is typically annual and time is in years.
What is compound interest and how does it differ from simple interest?: Compound interest is calculated on the original principal plus accumulated interest from previous periods. Formula: Final amount = Principal × (1 + rate/100)^time. Unlike simple interest, compound interest grows exponentially because interest is earned on interest.
What is a percentage multiplier and how is it used?: A percentage multiplier is a decimal that represents a percentage change. To increase by r%, multiply by (1 + r/100). To decrease by r%, multiply by (1 - r/100). Example: A 15% increase uses multiplier 1.15; a 15% decrease uses multiplier 0.85.
How do you calculate a percentage of a quantity using a multiplier?: Convert the percentage to a decimal multiplier by dividing by 100, then multiply by the quantity. Example: 23% of 150 = 0.23 × 150 = 34.5. Alternatively, use the fraction equivalent.