This cheat sheet covers the main number and ratio skills needed for GCSE Foundation Tier math. Students use these skills in arithmetic, problem solving, money questions, measures, maps, and real life contexts. It gives quick reminders for methods that are easy to mix up, such as percentage change, ratio sharing, and unit conversion.
It is designed as a clear reference for revision, homework, and exam practice.
The most important ideas are choosing the right operation, writing values in equivalent forms, and keeping units consistent. Fractions, decimals, and percentages are connected by division and multiplication by . Ratio compares parts, while proportion compares how quantities change together.
Standard form, estimation, and bounds help students handle very large or very small numbers and judge whether answers are reasonable.
Key Facts
- To convert a fraction to a percentage, calculate .
- To find of an amount , calculate .
- To increase an amount by , multiply by .
- To decrease an amount by , multiply by .
- In standard form, a number is written as , where and is an integer.
- To share an amount in the ratio , one part is , so the shares are and .
- For direct proportion, if is directly proportional to , then for a constant .
- Speed, distance, and time are linked by .
Vocabulary
- Place value
- Place value is the value of a digit based on its position in a number, such as ones, tenths, or thousands.
- Standard form
- Standard form writes a number as , where and is an integer.
- Percentage multiplier
- A percentage multiplier is the decimal factor used to increase or decrease an amount by a percentage.
- Ratio
- A ratio compares quantities by showing how many parts of one quantity match parts of another quantity.
- Direct proportion
- Direct proportion means two quantities increase or decrease at the same rate, so their ratio stays constant.
- Unit rate
- A unit rate compares a quantity to one unit of another quantity, such as miles per hour or cost per kilogram.
Common Mistakes to Avoid
- Adding percentages directly to the original number without a multiplier is wrong because increase means multiplying by , not adding unless the original amount is .
- Sharing a ratio by dividing by the number of ratio parts incorrectly is wrong because the total parts in are , not just .
- Moving the decimal the wrong way in standard form is wrong because multiplying by moves the decimal right when is positive and left when is negative.
- Comparing fractions using only the numerators is wrong because the denominator changes the size of each part, so use a common denominator or convert to decimals.
- Using mixed units in a rate calculation is wrong because formulas such as require consistent units before calculating.
Practice Questions
- 1 Write as a fraction in its simplest form and as a percentage.
- 2 Increase by using a percentage multiplier.
- 3 Share in the ratio .
- 4 A recipe uses flour and sugar in the ratio . Explain why doubling both amounts keeps the taste the same.
Understanding GCSE Foundation Tier Number and Ratio Reference
Place value is the structure behind every calculation. A digit has a different value depending on its position, so zeroes cannot be ignored. This matters when rounding, comparing decimals, and working with measures.
Rounding should use the digit immediately to the right of the place requested. Bounds go further than rounding. If a length is given as 6.2 centimetres to the nearest tenth, the true length is from 6.15 centimetres up to, but not including, 6.25 centimetres.
Bounds are useful when an answer comes from rounded measurements. Standard form uses powers of ten to show how far a decimal point has moved.
A positive power represents a large number and a negative power represents a small decimal. Keep the first number between one and ten before checking the power.
Fractions, decimals, and percentages describe the same kind of value, but each form is useful in different situations. Fractions are often best for exact answers, especially in algebra or probability. Decimals are practical for money and calculator work.
Percentages make comparisons easier when totals differ. A common mistake is to treat a percentage change as a fixed number of points. A ten percent rise followed by a ten percent fall does not return to the starting amount.
The second change is based on a different amount. This is important in sale prices, interest, depreciation, and population changes. For repeated changes, use the same multiplier each time rather than adding percentages together.
Ratios only make sense when the quantities are comparable. A paint mixture of two parts blue to three parts white describes parts, not actual litres. Before sharing or scaling, simplify the ratio if possible and find the total number of parts.
In a recipe, doubling every ingredient keeps the taste the same because every part is scaled by the same factor. In a map, a scale converts a drawing distance into a real distance, so units need careful attention. Direct proportion means that multiplying one quantity by a factor multiplies the other by that same factor.
Its graph is a straight line through the origin. If the graph does not pass through the origin, it is not direct proportion. Rates combine different units.
Speed may be in kilometres per hour, while a question gives time in minutes. Convert one unit before calculating.
Good number work includes checking whether an answer fits the situation. An estimate made by rounding numbers to one significant figure can reveal a misplaced decimal point or an incorrect calculator entry. Use estimation before a detailed calculation when possible.
Read wording closely for phrases such as percentage of, percentage increase, per, each, and in the ratio. They signal different methods. Show enough working to make your choices clear, particularly when converting units or finding a scale factor.
In exam questions, give exact values when requested, then round only at the end if a degree of accuracy is stated. Early rounding can change a final answer, especially in multi-step problems.