Solving Everyday Problems with Ratios and Proportions

Good to see you again. In the previous lesson, you used percentages to compare a part with an original whole. Ratios use the same underlying idea of comparison, but they are especially useful when two quantities must stay in a fixed relationship: ingredients in a recipe, people in groups, distance and time, or price and quantity.

This lesson focuses on recognising whether a problem involves a ratio, a rate, or a proportion, then choosing a reliable way to calculate an unknown value. By the end, you should be able to solve common exam-style everyday problems and explain why your answer makes sense.

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Three connected ideas: ratios, rates, and proportions

A ratio compares two quantities. For example, if there are $2$ red counters and $3$ blue counters, the ratio of red to blue is:

$$2:3$$

Read this as “$2$ to $3$” or “$2$ for every $3$.”

The order matters. The ratio of blue to red is:

$$3:2$$

These are different statements, just as “$3$ blue counters for every $2$ red counters” is different from the reverse.

Ratios can be simplified just like fractions. If a class has $12$ students who walk to school and $18$ who cycle, the walking-to-cycling ratio is:

$$12:18$$

Both parts divide by $6$:

$$12:18=2:3$$

This does not mean there are only $2$ walkers and $3$ cyclists. It describes the relationship: for every $2$ walkers, there are $3$ cyclists.

Part-to-part and part-to-whole ratios

Be precise about what is being compared.

If a bag contains $2$ red counters and $3$ blue counters:

• red to blue is a part-to-part ratio: $2:3$;
• red to all counters is a part-to-whole ratio: $2:5$, because there are $2+3=5$ counters altogether.

A common error is to use $2:3$ when the question asks for red counters out of the total. Always check the wording: does it ask for one group compared with another group, or one group compared with the whole?

A rate is a ratio comparing quantities with different units. Examples include:

$$60\text{ km per hour}$$ $$4\text{ dollars per kilogram}$$ $$15\text{ pages per minute}$$

A rate does not have to involve time, although many rates do. Price per kilogram is a rate because it compares money with mass.

Finally, a proportion says that two ratios are equal. For example:

$$\frac{2}{3}=\frac{10}{15}$$

Both ratios describe the same relationship. Proportions allow you to scale a known ratio up or down while keeping it equivalent.

In the hamster example, the number of hamsters changes from $1$ to $5$. That is a scale factor of $5$, so the food also changes from $60$ grams to:

$$60\times 5=300$$

grams.

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Scaling ratios without losing the relationship

The simplest proportional problems can be solved by finding the scale factor.

Suppose a recipe uses $3$ cups of flour for every $2$ eggs. You want to use $15$ cups of flour.

The flour amount has changed from $3$ to $15$:

$$15÷3=5$$

So the recipe is being made $5$ times as large. Multiply the eggs by the same factor:

$$2\times 5=10$$

You need:

$$10\text{ eggs}$$

The important rule is not “multiply by five” on its own. The real rule is:

When a relationship is proportional, both matching quantities must be scaled by the same factor.

You can also find the answer by working out the amount for one unit. In the recipe:

$$3÷2=1.5$$

So each egg needs $1.5$ cups of flour. For $15$ cups of flour:

$$15÷1.5=10$$

eggs.

Both methods are valid. Use a scale factor when the multiplication or division is obvious; use a per-unit amount when it is clearer.

Sometimes the scale factor is not easy to spot. Then write a proportion, keeping the units in the same positions:

$$\frac{\text{flour}}{\text{eggs}}=\frac{\text{flour}}{\text{eggs}}$$

For the recipe:

$$\frac{3}{2}=\frac{15}{x}$$

Here, $x$ is the number of eggs. Cross-multiplying gives:

$$3x=2\times 15$$ $$3x=30$$ $$x=10$$

The layout works because flour is on top in both fractions and eggs are on the bottom in both fractions. If you swap the order in only one fraction, the calculation will be wrong.

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Rates and unit rates: finding the amount for one

A unit rate is a rate written for one unit. It makes comparisons much easier.

For example, a cyclist travels $72$ kilometres in $3$ hours. The unit rate is:

$$72÷3=24$$

So the cyclist’s average speed is:

$$24\text{ km per hour}$$

If the cyclist continues at that same rate for $5$ hours, the distance is:

$$24\times 5=120$$

So the cyclist travels:

$$120\text{ km}$$

Notice the condition at the same rate. You may use proportional reasoning only when the relationship stays constant. A journey with long stops, changing speeds, or a taxi fare with a starting charge may not be directly proportional.

Unit price and best value

Unit rates are especially useful when comparing deals. The lowest overall price is not always the best value, because the amounts may be different.

Imagine two rice offers:

Offer	Total price	Mass	Unit price
A	$5.70$ dollars	$1.5$ kg	$5.70÷1.5=3.80$ dollars per kg
B	$7.20$ dollars	$2$ kg	$7.20÷2=3.60$ dollars per kg

Offer B costs more at the checkout, but it is cheaper per kilogram. Therefore, it is better value if you want the same product and will use the full amount.

When comparing deals:

1. Make sure you are comparing the same unit: for example, dollars per kilogram for both products.
2. Divide the total cost by the quantity.
3. Choose the lower unit price.

The same logic works for cost per ticket, pay per hour, fuel used per kilometre, or data per month.

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A reliable method for everyday proportional problems

In an exam, the context may look unfamiliar, but the mathematical decisions are usually the same. Use this routine.

1. Identify the quantities and their units.
   For example: people and meals, kilometres and hours, dollars and kilograms.

2. Identify what is fixed.
   Look for clues such as “for every,” “same recipe,” “at a constant speed,” or “per.”

3. Decide which tool fits.

   • Use a ratio to compare groups.
   • Use a rate to find an amount per one unit.
   • Use a proportion when a fixed relationship is scaled and one value is unknown.

4. Keep corresponding units aligned.
   If you write fractions, place the same type of quantity on top each time, or on the bottom each time.

5. Calculate and state the unit in the answer.
   A bare answer such as $40$ is incomplete if the question asks for $40$ teachers, $40$ minutes, or $40$ dollars.

6. Check whether the answer is reasonable.
   If the number of people doubles, the amount of food should double in a direct proportion. If a larger pack has the lower unit price, its price per unit should be smaller, not larger.

Worked example: servings and ingredients

A canteen uses $1.2$ kg of pasta to make $8$ servings. How much pasta is needed for $30$ servings, assuming serving size stays the same?

First find the pasta per serving:

$$1.2÷8=0.15$$

So each serving needs:

$$0.15\text{ kg}$$

For $30$ servings:

$$0.15\times 30=4.5$$

The canteen needs:

$$4.5\text{ kg of pasta}$$

A quick check supports the answer. Since $30$ servings is almost four times $8$ servings, the required pasta should be almost four times $1.2$ kg. The answer, $4.5$ kg, fits that expectation.

Common mistakes to avoid

• Reversing a ratio.
  “Boys to girls” is not the same as “girls to boys.”

• Adding to a ratio instead of scaling it.
  A ratio of $2:3$ scaled by $4$ becomes $8:12$, not $6:7$.

• Using values that do not match.
  If $6$ tickets cost $48$ dollars, divide $48$ by $6$, not by a number from a different offer.

• Comparing total prices rather than unit prices.
  A larger package often costs more overall but may still be better value.

• Forgetting the constant-rate assumption.
  Proportional reasoning is valid only when the relationship remains fixed.

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The key idea is that ratios describe a relationship, rates describe a comparison involving different units, and proportions preserve the same relationship while quantities change.

For a direct proportion, you can either scale both quantities by the same factor, find a unit rate, or write equivalent ratios:

$$\frac{a}{b}=\frac{c}{d}$$

Keep the units in the correct positions, include units in your final answer, and check whether the size of the answer fits the situation.

Next lesson, you will use similar careful thinking with metric unit conversions, length, area, volume, and elapsed time.