Mastering Ratio and Proportion for Your Mathematics Exam

What the examiner is testing

The examiner is testing your ability to scale quantities up or down proportionally and to divide a total into parts according to a given ratio. Marks are awarded for correctly setting up the proportion or ratio and for accurate calculation of the unknown values.

The method

1. Identify the relationship: Determine if the problem involves a ratio (comparing parts) or proportion (scaling quantities).
2. Establish known values: Write down the given ratio or the proportional relationship using the units provided.
3. Find the "multiplier" or "value per part":
   • For ratio problems with a total, sum the parts of the ratio and divide the total by this sum.
   • For proportion problems, divide the known output by the known input to find the scaling factor.
4. Apply the multiplier/value per part: Multiply each part of the ratio or the new input by the value found in step 3 to find the unknown quantities.
5. Check for consistency: Ensure your final answers maintain the original ratio or proportional relationship.

Worked example

A recipe requires flour and sugar in the ratio \(3:2\). If you use \(180 \text{ g}\) of flour, how much sugar do you need?

$$ \text{Flour} : \text{Sugar} = 3 : 2 $$

We know the amount of flour is \(180 \text{ g}\).
The ratio part for flour is \(3\).

$$ 1 \text{ part} = \frac{180 \text{ g}}{3} $$

$$ 1 \text{ part} = 60 \text{ g} $$

Now, find the amount of sugar. The ratio part for sugar is \(2\).

$$ \text{Sugar} = 2 \times 60 \text{ g} $$

$$ \text{Sugar} = 120 \text{ g} $$

Sanity check: The ratio of flour to sugar is \(180:120\). Dividing both by \(60\) gives \(3:2\), which matches the original ratio.

Worked example: a harder one

Share \(£120\) between Amy, Ben, and Chloe in the ratio \(2:3:5\). Amy then gives \(25\%\) of her share to Ben. How much money does each person have now?

First, try to simply apply the \(25\%\) reduction to Amy's initial share. This fails because the \(25\%\) is after the initial distribution, not part of the original ratio calculation.

Step 1: Find the total number of parts.
$$ 2 + 3 + 5 = 10 \text{ parts} $$

Step 2: Find the value of one part.
$$ 1 \text{ part} = \frac{£120}{10} $$

$$ 1 \text{ part} = £12 $$

Step 3: Calculate initial shares.
Amy's share: \(2 \times £12 = £24\)
Ben's share: \(3 \times £12 = £36\)
Chloe's share: \(5 \times £12 = £60\)

Check total: \(£24 + £36 + £60 = £120\). This is correct.

Step 4: Amy gives \(25\%\) of her share to Ben.
Calculate \(25\%\) of Amy's share:
$$ 0.25 \times £24 = £6 $$

Step 5: Adjust shares.
Amy's new share: \(£24 - £6 = £18\)
Ben's new share: \(£36 + £6 = £42\)
Chloe's share remains: \(£60\)

Final check: \(£18 + £42 + £60 = £120\). The total money is still the same.

Practice

1. Simplify the ratio \(35:21\).
2. A map has a scale of \(1:50,000\). If a road is \(4 \text{ cm}\) long on the map, what is its actual length in kilometres?
3. Sarah and Tom share some money in the ratio \(5:3\). If Sarah receives \(£40\) more than Tom, how much money do they have in total?
4. In a bag, there are red, blue, and green counters. The ratio of red to blue counters is \(3:4\). The ratio of blue to green counters is \(6:5\). If there are \(30\) green counters, how many red counters are there?

Answers:

1. \(5:3\)
2. \(2 \text{ km}\)
3. \(£160\)

4. Working for Q4:
   We have two ratios: \(R:B = 3:4\) and \(B:G = 6:5\).
   To combine these, we need a common value for B. The lowest common multiple of \(4\) and \(6\) is \(12\).
   Multiply the first ratio by \(3\): \(R:B = (3 \times 3) : (4 \times 3) = 9:12\)
   Multiply the second ratio by \(2\): \(B:G = (6 \times 2) : (5 \times 2) = 12:10\)
   Now we can combine them: \(R:B:G = 9:12:10\)

   We are given that there are \(30\) green counters.
   The ratio part for green is \(10\).
   $$ 1 \text{ part} = \frac{30 \text{ counters}}{10} $$
   $$ 1 \text{ part} = 3 \text{ counters} $$
   We need to find the number of red counters. The ratio part for red is \(9\).
   $$ \text{Red counters} = 9 \times 3 \text{ counters} $$
   $$ \text{Red counters} = 27 \text{ counters} $$

The three mistakes that lose marks

1. Adding ratio parts when they represent quantities, not a total: If a question states "The ratio of apples to oranges is \(2:3\), and there are \(10\) apples," a common mistake is to say \(2+3=5\) parts, so \(10/5=2\) per part. This is wrong because \(10\) is the number of apples, not the total fruit. The correct approach is \(2\) parts = \(10\) apples, so \(1\) part = \(5\) apples.
2. Incorrectly combining ratios with different common terms: When combining \(A:B\) and \(B:C\), students sometimes just write \(A:B:C\) as \(A:B:C\) without making the \(B\) values equal. For example, if \(A:B = 1:2\) and \(B:C = 3:4\), simply writing \(1:2:4\) is incorrect. The \(B\) values must be made the same (e.g., \(A:B = 3:6\) and \(B:C = 6:8\), so \(A:B:C = 3:6:8\)).
3. Confusing direct and inverse proportion: Assuming all proportion problems are direct. For example, if "5 people take 10 hours to paint a fence," assuming "10 people take 20 hours" (direct proportion) is incorrect. More people means less time (inverse proportion), so 10 people would take 5 hours.

30-second recap

Ratio compares parts of a whole or different quantities. Proportion describes how quantities scale relative to each other. Always find the value of "one part" or the "scaling factor" first, then apply it to find the unknowns. Pay close attention to whether the problem involves a total or individual quantities.