Ratios, equivalent proportions and mixtures · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| part-to-whole ratio/pɑːt tə həʊl ˈreɪʃɪəʊ/ | 部分与整体的比 | bù fèn yǔ zhěng tǐ de bǐ |
A recipe uses concentrate and water in the ratio 2:5. A 700 ml batch and a doubled batch should taste the same, although their volumes differ.
- A recipe uses concentrate and water in the ratio 2:5. A 700 ml batch and a doubled batch should taste the same, although their volumes differ.
- This lesson studies part-to-whole ratio 部分与整体的比: A comparison of one share with the combined quantity.
Choose the mathematical structure
- Put quantities in the same units before simplifying a ratio. Divide every part by the same nonzero factor. For a:b the whole has a+b parts; the first share is a/(a+b) of the whole but a/b of the other share. Equivalent ratios have the same multiplicative relationship: a/b=c/d gives ad=bc when denominators are nonzero.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines part-to-whole ratio?
A comparison of one share with the combined quantity.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For 700 ml at 2:5, each part is 100 ml: concentrate 200 ml and water 500 ml. Concentrate:whole=2:7, while concentrate/water=2/5. Water/concentrate=5/2, a fraction greater than 1. Doubling both gives 400:1000=2:5. If concentrate is x and water y, the recipe requires y=(5/2)x, a straight line through the origin. For 300 ml concentrate, water is 750 ml and the total is 1050 ml. A batch with 200 ml concentrate and 600 ml water has ratio 1:3 and is weaker. To simplify 1.5 litres:500 ml, first write 1500:500=3:1.
Ratios, equivalent proportions and mixtures
Put quantities in the same units before simplifying a ratio
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find concentrate in a 700 ml mixture at concentrate:water=2:5.
700/(2+5)×2=200 ml.
Test a tempting shortcut
- Adding the same amount to both shares generally changes the ratio. Do not divide by 2+7 when the stated ratio is already part:whole 2:7; the whole is seven parts. A concentration fraction uses total volume as denominator.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Adding 100 ml to both shares always preserves a mixture ratio. This claim is false. Explain which definition or assumption it violates.
Find the numerator of water/concentrate in lowest terms.
Water/concentrate=500/200=5/2.
Adding 100 ml to both shares always preserves a mixture ratio.
Adding the same amount to both shares generally changes the ratio. Do not divide by 2+7 when the stated ratio is already part:whole 2:7; the whole is seven parts. A concentration fraction uses total volume as denominator.
Interpret a new situation
- AQA R3–R8 connects ratio notation, fractions, equivalent proportions and linear functions. Check that both shares sum to the stated total, and name which quantity is compared with which. Use a ratio table to scale a mixture without assuming a fixed additive difference.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For y=(5/2)x, find y when x=300.
Multiply the concentrate by 5/2: 300×2.5=750.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A comparison of one share with the combined quantity. Choose the relationship, show the method, check its assumptions and interpret the result.