Converting Fractions, Decimals, and Percentages in Context

Hello, and welcome to the first lesson of your Numeracy Exam Sprint. This module builds the number skills that appear across many exam questions: interpreting information accurately, choosing the right calculation, and showing an answer in the form requested.

Fractions, decimals, and percentages are not three different quantities. They are three ways of writing the same proportion of a whole. In this lesson, you will learn to move confidently between them, recognise which form an exam question needs, and check that an answer makes sense in context.

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One amount, three representations

A fraction describes equal parts of a whole:

$$\frac{\text{number of parts chosen}}{\text{total equal parts}}$$

In $\frac{3}{4}$, the numerator $3$ tells you how many parts are being considered, and the denominator $4$ tells you the total number of equal parts.

A decimal uses place value:

• $0.3$ means three tenths, or $\frac{3}{10}$
• $0.03$ means three hundredths, or $\frac{3}{100}$
• $0.003$ means three thousandths, or $\frac{3}{1000}$

A percentage means “out of one hundred”:

$$75\%=\frac{75}{100}$$

So these all represent exactly the same amount:

$$\frac{3}{4}=0.75=75\%$$

Imagine a test with $24$ questions where a student answers $18$ correctly. The score can be written in several equivalent forms:

$$\frac{18}{24}=\frac{3}{4}=0.75=75\%$$

The value has not changed; only its representation has changed. In an exam, the question determines the form you should give:

• “What fraction of the questions?” calls for a fraction.
• “Write the probability as a decimal” calls for a decimal.
• “What percentage was correct?” calls for a percentage.

Before converting, always identify the whole. A fraction or percentage is meaningful only in relation to a total.

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The central idea: tenths, hundredths, and “per hundred”

The easiest conversions come from place value.

$$\frac{7}{10}=0.7=70\%$$ $$\frac{9}{100}=0.09=9\%$$ $$\frac{125}{1000}=0.125=12.5\%$$

Notice the important difference between $0.5$ and $0.05$:

$$0.5=\frac{5}{10}=\frac{50}{100}=50\%$$ $$0.05=\frac{5}{100}=5\%$$

A zero can change the value dramatically. In an exam, writing $0.5$ when the correct value is $0.05$ makes an answer ten times too large.

Here is a visual example of turning a fraction into a decimal by making its denominator $100$.

Because $25\times 4=100$, we multiply both parts of the fraction by $4$:

$$\frac{21}{25}=\frac{21\times 4}{25\times 4}=\frac{84}{100}$$

Once the denominator is $100$, the other forms are immediate:

$$\frac{84}{100}=0.84=84\%$$

Multiplying the numerator and denominator by the same number does not change the fraction’s value. It simply changes its name into a more useful equivalent fraction.

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A reliable conversion toolkit

Rather than memorising disconnected rules, use the fact that every percentage is a fraction out of $100$.

Starting form	Method	Example
Fraction to decimal	Divide numerator by denominator, or make a denominator of $10$, $100$, or $1000$	$\frac{3}{4}=3÷4=0.75$
Decimal to fraction	Use place value, then simplify	$0.36=\frac{36}{100}=\frac{9}{25}$
Decimal to percentage	Multiply by $100$	$0.36=36\%$
Percentage to decimal	Divide by $100$	$36\%=0.36$
Percentage to fraction	Write it over $100$, then simplify	$36\%=\frac{36}{100}=\frac{9}{25}$
Fraction to percentage	Make the denominator $100$, or convert to a decimal then multiply by $100$	$\frac{3}{8}=0.375=37.5\%$

Fractions to decimals

If the denominator is already $10$, $100$, or $1000$, use place value directly:

$$\frac{6}{10}=0.6$$ $$\frac{17}{100}=0.17$$ $$\frac{4}{1000}=0.004$$

If you can turn the denominator into $100$, that is often quick. For example:

$$\frac{13}{25}=\frac{52}{100}=0.52$$

For other denominators, divide the numerator by the denominator:

$$\frac{5}{8}=5÷8=0.625$$

Some fractions produce decimals that continue forever:

$$\frac{1}{3}=0.333\dots$$

In a question requiring a decimal or percentage, follow any rounding instruction. If none is given, use the level of accuracy appropriate to the context; for example, money is normally rounded to two decimal places.

Decimals to fractions

Read the decimal using its place value, then simplify.

$$0.4=\frac{4}{10}=\frac{2}{5}$$ $$0.38=\frac{38}{100}=\frac{19}{50}$$ $$0.125=\frac{125}{1000}=\frac{1}{8}$$

The denominator depends on the number of decimal places:

• one decimal place: denominator $10$
• two decimal places: denominator $100$
• three decimal places: denominator $1000$

Always simplify unless the question specifically asks for a fraction with a particular denominator.

Decimals and percentages

Moving between decimals and percentages changes the scale between “out of $1$” and “out of $100$.”

To turn a decimal into a percentage, multiply by $100$:

$$0.7\times 100=70$$ $$0.7=70\%$$

To turn a percentage into a decimal, divide by $100$:

$$7\%=\frac{7}{100}=0.07$$

Writing a leading zero in a decimal is good exam practice:

$$8\%=0.08$$

not $.08$.

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Converting in context

In exam questions, the numbers may appear in a recipe, a survey, a score, a discount, or a probability statement. The calculation is the same, but the meaning of the answer matters.

Example 1: A score

A student answers $21$ questions correctly out of $25$.

The fraction correct is:

$$\frac{21}{25}$$

Make the denominator $100$:

$$\frac{21}{25}=\frac{84}{100}$$

Therefore:

$$\frac{21}{25}=0.84=84\%$$

A score of $84\%$ is sensible because $21$ is a little less than $25$, so the answer should be a little less than $100\%$.

Example 2: A portion of a recipe

A recipe uses $\frac{3}{8}$ of a kilogram of flour.

Using division:

$$3÷8=0.375$$

So the recipe uses $0.375$ kg, which is:

$$0.375\times 100\%=37.5\%$$

of one kilogram.

The decimal form is useful if the question asks for a measurement in kilograms. The percentage form is useful if the question asks for a comparison with a whole kilogram.

Example 3: A survey result

A survey states that $15\%$ of respondents chose a particular option.

As a decimal:

$$15\%=\frac{15}{100}=0.15$$

As a simplified fraction:

$$\frac{15}{100}=\frac{3}{20}$$

Thus:

$$15\%=0.15=\frac{3}{20}$$

The survey does not mean $15$ people chose the option unless the total number of respondents was $100$. It means $15$ out of every $100$, proportionally.

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Exam checks that catch common mistakes

A few quick checks can prevent lost marks.

Check the size

For a proper fraction, the numerator is smaller than the denominator. Its decimal should be less than $1$, and its percentage should be less than $100\%$.

For example:

$$\frac{7}{8}=0.875=87.5\%$$

These values are all below one whole, so they agree.

If you wrote $875\%$, the percentage would mean more than eight wholes. That cannot match $\frac{7}{8}$.

Keep equivalent fractions balanced

To create an equivalent fraction, multiply or divide the numerator and denominator by the same number.

Correct:

$$\frac{3}{5}=\frac{3\times 20}{5\times 20}=\frac{60}{100}=60\%$$

Incorrectly changing only one part of a fraction changes its value.

Do not confuse a decimal with its percentage

$$0.62=62\%$$

but:

$$0.62\%=0.0062$$

The percent sign is not decoration. It tells you the number is being compared with $100$.

Match the requested answer form

If an exam asks for a fraction in simplest form, do not stop at:

$$\frac{40}{100}$$

Simplify it:

$$\frac{40}{100}=\frac{2}{5}$$

If it asks for a percentage, include the percent sign:

$$\frac{2}{5}=40\%$$

A mathematically related answer can still lose marks if it does not answer the form requested.

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A short method for exam questions

When you see a conversion question, use this routine:

1. Underline the requested form: fraction, decimal, or percentage.
2. Identify the starting form and the whole being described.
3. Choose the most direct method:
   • use place value for denominators of $10$, $100$, or $1000$;
   • make the denominator $100$ when possible;
   • otherwise divide numerator by denominator.
4. Simplify fractions if required.
5. Check size and context: a result should be plausible for the quantity described.
6. Write the correct label or symbol, such as $\%$, kg, or “of the questions.”

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You now have one central connection to rely on:

$$\text{percentage}=\text{fraction out of }100$$

Fractions describe parts of a whole, decimals express those parts through place value, and percentages express them per hundred. Converting is not a collection of unrelated tricks; it is a change of representation while keeping the amount the same.

Next, you will use these representations to calculate percentage amounts, percentage increases and decreases, and discounts—skills that build directly on writing percentages accurately.