Grades, Fractions and Proportions: Everyday School Maths Made Clear
A few small ideas appear again and again in school, cooking, shopping and work: weighted averages, simplest fractions and proportions. Once you see the pattern behind each one, the calculation takes seconds and the result is easy to check.
Weighted grades
When some items count more than others, multiply each score by its weight, add the results and divide by the total weight. Scores of 85, 90 and 78 with weights of 30, 20 and 50 behave like this:
- Multiply each score by its weight: 85 × 30 = 2,550, 90 × 20 = 1,800 and 78 × 50 = 3,900.
- Add the weighted scores: 2,550 + 1,800 + 3,900 = 8,250.
- Divide by the total weight, 100: 8,250 ÷ 100 = 82.5.
- The plain average would be 84.3, so the heavier final item pulls the result down by 1.8 points.
The Weighted Grade Calculator also answers the reverse question: what score is needed on a final exam to reach a target. With an average of 82.5, a final worth 30% and a target of 80, the score needed is (80 − 82.5 × 0.7) ÷ 0.3 = 74.17.
Simplifying fractions
Divide the top and the bottom by their greatest common divisor. For 18/24 the divisor is 6, so the fraction becomes 3/4. The Fraction Simplifier returns the simplified form together with the decimal 0.75 and the percentage 75%, which is often the most useful version in a shop or a recipe.
Adding fractions needs a common denominator. One third plus one quarter is 4/12 + 3/12 = 7/12, about 0.583. Adding the tops and the bottoms would give 2/7, which is both wrong and smaller than either part. The GCD & LCM Calculator finds the divisor or the common denominator when the numbers are larger.
The rule of three
A proportion compares two ratios. In a direct proportion, more of one thing means more of the other. If 3 kg cost 12, then 5 kg cost 12 × 5 ÷ 3 = 20. The unit price is 4 per kilogram, and 5 × 4 confirms it.
In an inverse proportion, more of one thing means less of the other. If 4 workers need 6 days, the job is 24 worker-days; 3 workers need 24 ÷ 3 = 8 days. The question to ask is always whether the second quantity should rise or fall, because that decides whether you divide or multiply.
- Write the known pair and the known value beside the unknown.
- Decide whether the relationship is direct or inverse.
- For a direct proportion, multiply across and divide by the remaining figure.
- For an inverse proportion, multiply the pair and divide by the remaining figure.
- Check the direction: more workers should shorten the job, not lengthen it.
The Rule of Three Calculator handles both types and names the type in the result.
Fractions, decimals and percentages
These three forms describe the same number, and moving between them is mostly a matter of division or multiplication by 100.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
Eighths are worth learning because they appear on rulers, in recipes and in prices. Half of 1/4 is 1/8, and half of 3/4 is 3/8, so the values follow from each other.
Comparing pack sizes
Proportions decide which pack is better value. Divide the price by the quantity to get a price per unit, then compare like with like.
| Pack | Price | Per kg |
|---|---|---|
| 500 g | 2.10 | 4.20 |
| 1 kg | 3.90 | 3.90 |
| 2 kg | 7.20 | 3.60 |
| 3 kg | 10.20 | 3.40 |
The largest pack is not automatically the cheapest per kilogram, and a special offer can change the order. Convert the units first: 500 g is 0.5 kg, so 2.10 ÷ 0.5 = 4.20. The Percentage Calculator is useful when a discount is quoted as a percentage rather than a price.
Check the answer
Three quick checks catch most errors. Does the result point in the right direction? If more workers make the job take longer, the proportion was set up as the wrong type. Is the size plausible? A weighted average must lie between the lowest and the highest score, and a simplified fraction must have the same value as the original. Do the parts add up? A percentage breakdown of a single total should sum to 100%.
Mistakes that cost marks and money
- Adding fractions across. Use a common denominator instead of adding the tops and the bottoms.
- Forgetting the weights. A plain average of scores with different weights answers a different question.
- Using the wrong proportion. Decide whether the quantities rise together before multiplying or dividing.
- Comparing packs in different units. Convert to the same unit before comparing prices.
- Rounding before the end. Keep the exact fraction or a few decimals until the final line.
Frequently asked questions
Scores of 85, 90 and 78 with weights 30, 20 and 50: what is the average?
82.5, from (2,550 + 1,800 + 3,900) ÷ 100. The plain average would be 84.3, but it ignores the weights.
I have an average of 82.5 and the final is worth 30%. What do I need for 80?
74.17, from (80 − 82.5 × 0.7) ÷ 0.3. The final counts for 30%, so the existing average counts for 70%.
Simplify 18/24.
3/4, dividing both numbers by 6. As a decimal it is 0.75, and as a percentage 75%.
4 workers take 6 days. How long do 3 workers take?
8 days, because the job is 24 worker-days and 24 ÷ 3 = 8. This is an inverse proportion: fewer workers means more days.
- Weighted Grade Calculator – Weighted average of scores, and the final exam score you need.
- Fraction Simplifier – Simplify a fraction and see it as a decimal and a percent.
- Rule of Three Calculator – Solve direct and inverse proportion problems.
- GCD & LCM Calculator – Find the greatest common divisor and least common multiple.
- Percentage Calculator – Calculate a percentage of any number.
More guides
- How to Calculate Percentages: 3 Methods and the Mistakes to Avoid
- Understanding Compound Interest: Why Time Matters More Than Amount
- How Loan Payments Work: Principal, Interest and the Cost of Time
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This guide is for general information and is not financial, tax, legal or medical advice. See our disclaimer.