Number & Algebra
Number & Numeration: Fractions, Decimals, Percentages, Approximation, Indices & Standard Form
JSS 1 · JSS 2 · JSS 3
- ✓By the end of this lesson students will be able to perform operations on fractions, decimals, and percentages.
- ✓By the end of this lesson students will be able to convert between fractions, decimals, and percentages.
- ✓By the end of this lesson students will be able to approximate numbers to a given number of decimal places or significant figures.
- ✓By the end of this lesson students will be able to apply the basic laws of indices to simplify expressions.
- ✓By the end of this lesson students will be able to express numbers in standard form and convert from standard form to ordinary numbers.
Key concepts
A fraction represents a part of a whole. It is written as a ratio of two numbers, the numerator (top number) and the denominator (bottom number). The denominator indicates the total number of equal parts, and the numerator indicates how many of those parts are being considered. Types include proper fractions (numerator < denominator), improper fractions (numerator ≥ denominator), and mixed numbers (a whole number and a proper fraction).
Decimals are another way of representing fractions where the denominator is a power of ten (e.g., 10, 100, 1000). The decimal point separates the whole number part from the fractional part. Each digit after the decimal point represents a fraction with a denominator that is a power of ten (e.g., 0.1 = 1/10, 0.01 = 1/100).
A percentage is a way of expressing a number as a fraction of 100. The word 'percent' means 'per hundred' or 'out of one hundred'. It is denoted by the symbol '%'. Percentages are used to represent parts of a whole, often for comparison or to show proportions.
Approximation is the process of finding a value that is close enough to the exact value for a specific purpose. This often involves rounding numbers to a certain number of decimal places or significant figures. Rounding rules: If the digit to be dropped is 5 or greater, round up the preceding digit. If it is less than 5, keep the preceding digit as it is.
Indices (plural of index) or exponents are a shorthand way of writing repeated multiplication of the same number. In the expression a^n, 'a' is called the base and 'n' is called the index or exponent. It means 'a' is multiplied by itself 'n' times.
These are rules that simplify calculations involving indices:
Standard form is a convenient way to write very large or very small numbers. A number is in standard form when it is written as A × 10^n, where 'A' is a number between 1 and 10 (i.e., 1 ≤ A < 10) and 'n' is an integer (positive for large numbers, negative for small numbers).
Key facts to remember
- 1To add or subtract fractions, find the LCM of the denominators.
- 2To divide by a fraction, multiply by its reciprocal.
- 3To convert a decimal to a percentage, multiply by 100%. To convert a percentage to a decimal, divide by 100.
- 4When rounding, if the digit to be dropped is 5 or more, round up the preceding digit.
- 5Significant figures start from the first non-zero digit.
- 6Any non-zero number raised to the power of zero is 1 (e.g., a^0 = 1).
- 7In standard form, the number 'A' must be between 1 and 10 (1 ≤ A < 10).
- 8A positive exponent in standard form means a large number, a negative exponent means a small number.
Worked examples
Example 1
Evaluate (3/4 + 1/3) ÷ 1.25, expressing your answer as a percentage to 1 decimal place.
Answer
86.7%
Always show your working clearly, especially when converting between forms and rounding.
Example 2
Simplify (2^3 × 2^5) / 2^4 and express 0.000078 in standard form.
Answer
16 and 7.8 × 10^-5
Remember that moving the decimal point to the right for a small number results in a negative exponent in standard form.
Example 3
A student scored 45 out of 60 in a Mathematics test. What is her score as a percentage? Round your answer to 2 significant figures.
Answer
75%
Be careful with rounding. Sometimes a number might already be at the required precision.
Common mistakes
- ✗Adding or subtracting fractions without finding a common denominator.
- ✗Confusing the rules for multiplying and dividing indices (e.g., adding exponents for division).
- ✗Incorrectly placing the decimal point when converting to/from standard form, especially with negative exponents.
- ✗Rounding errors, particularly when dealing with significant figures and leading zeros (e.g., 0.005 rounded to 1 s.f. is 0.005, not 0.01).
- ✗Forgetting to multiply by 100 when converting a fraction or decimal to a percentage.
Exam tips
- ★Always show all steps of your working clearly, as marks are often awarded for method.
- ★Read the question carefully to ensure you answer exactly what is asked, especially regarding units and required precision (e.g., 'to 2 decimal places' or 'to 3 significant figures').
- ★Practise converting between fractions, decimals, and percentages regularly to build speed and accuracy.
- ★Memorise the basic laws of indices; they are fundamental to solving problems involving powers.
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