Challenging Fractions, Decimals, and Percentages Problems

Hello again. Last lesson, you solved number-property riddles by turning each clue into a mathematical condition and checking every answer. The same careful habit matters with fractions, decimals and percentages: they often describe the same amount in different forms.

This lesson is questions 13–20 of your hard Year 6 SATs challenge. You will compare values, calculate fractions and percentages of amounts, find what is left, and solve a reverse-percentage problem. These are SATs-style reasoning skills that will also be useful as you move into Year 7.

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One value, three ways to write it

A fraction, decimal and percentage can all name the same part of a whole.

Fraction	Decimal	Percentage
$\frac{1}{2}$	$0.5$	$50\%$
$\frac{1}{4}$	$0.25$	$25\%$
$\frac{3}{4}$	$0.75$	$75\%$
$\frac{1}{5}$	$0.2$	$20\%$
$\frac{2}{5}$	$0.4$	$40\%$
$\frac{3}{5}$	$0.6$	$60\%$
$\frac{4}{5}$	$0.8$	$80\%$
$\frac{1}{10}$	$0.1$	$10\%$

Remember that percent means “out of 100”:

$$35\%=\frac{35}{100}=0.35$$

For SATs, do not try to convert everything in every question. Choose the form that makes the calculation easiest:

• use decimals when comparing values or working with money;
• use fractions when finding a fraction of an amount or adding fractional parts;
• use percentages when the question gives a discount, survey result, or “out of 100” situation.

Here is one important comparison trap:

$$0.7=0.70$$ $$0.07=0.07$$

Although $7$ is greater than $0$, $0.7$ is greater because it has seven tenths, while $0.07$ has only seven hundredths. Adding a zero can make place value easier to see.

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Three reliable methods for hard questions

1. Make the units match before comparing

To order $\frac{5}{8}$, $0.62$, and $63\%$, change them into decimals:

$$\frac{5}{8}=0.625$$ $$63\%=0.63$$

Now compare:

$$0.62,0.625,0.63$$

The zeroes and equal decimal places help you compare accurately.

2. Find a fraction or percentage of an amount

The word of usually means multiply.

For a fraction of an amount, divide by the denominator first, then multiply by the numerator. For example:

$$\frac{3}{8}\text{ of }640$$

First find one eighth:

$$640÷8=80$$

Then find three eighths:

$$80\times 3=240$$

For a percentage, use useful chunks. To find $35\%$ of $780$:

$$10\%=78$$ $$30\%=234$$ $$5\%=39$$ $$35\%=234+39=273$$

3. Treat the whole as $1$, $100\%$, or a full amount

When a question asks what is left, start with the complete whole and subtract the known parts.

For example, if $\frac{5}{12}$ is blue and $0.35$ is red, convert the decimal to a fraction:

$$0.35=\frac{35}{100}=\frac{7}{20}$$

The denominators are different, so use a common denominator of $60$:

$$\frac{5}{12}=\frac{25}{60}$$ $$\frac{7}{20}=\frac{21}{60}$$

Together, the known parts are:

$$\frac{25}{60}+\frac{21}{60}=\frac{46}{60}$$

The part left is:

$$\frac{60}{60}-\frac{46}{60}=\frac{14}{60}=\frac{7}{30}$$

Do not add or subtract fractions until their denominators match.

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A special challenge: reverse percentages

A discount question often asks for the original price. This is different from finding a percentage of an amount.

Suppose a phone costs £132 after a $20\%$ reduction.

A reduction of $20\%$ means the buyer pays:

$$100\%-20\%=80\%$$

So £132 represents $80\%$ of the original price.

If $80\%$ is £132, then $10\%$ is:

$$\pounds 132÷8=\pounds 16.50$$

The original $100\%$ is ten lots of £16.50:

$$\pounds 16.50\times 10=\pounds 165$$

The original price was:

$$\boxed{\pounds 165}$$

A common mistake is to find $20\%$ of £132. That would use the sale price as though it were the original price, so it cannot work.

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Eight-question hard SATs challenge

Work carefully and show at least one line of working for every question. In reasoning papers, correct working can help you spot mistakes before you commit to an answer.

Question 13

Put these values in ascending order, from smallest to largest:

$$\frac{3}{5},0.62,\frac{5}{8},63\%$$

Question 14

Find the missing number.

$$\frac{2}{5}+\square =1.15$$

Give your answer as either a fraction in its simplest form or a decimal.

Question 15

A youth club has £640 to spend on an event.

• It spends $\frac{3}{8}$ of the money on equipment.
• It spends $25\%$ of the original money on transport.

How much money is left? What percentage of the original £640 is left?

Question 16

In a mural, $\frac{5}{12}$ is painted blue and $0.35$ is painted red. The rest is painted yellow.

What fraction of the mural is yellow? Give your answer in its simplest form.

Question 17

Find $35\%$ of $780\text{ kg}$.

Question 18

During a sale, a phone’s price is reduced by $20\%$. Jack pays £132.

What was the original price of the phone?

Question 19

A factory makes $3600$ sweets every hour.

• $\frac{5}{9}$ are lollipops.
• $20\%$ are gummy bears.
• The rest are chocolate bars.

How many chocolate bars are made each hour?

Question 20

Pippa starts with some money.

• She spends $\frac{1}{3}$ of it on a pencil case.
• She then spends $\frac{1}{2}$ of what she has left on pens.
• The pens cost £18.

How much money did Pippa have at the start?

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Self-marking and worked solutions

Question 13

Convert everything to decimals:

$$\frac{3}{5}=0.6$$ $$\frac{5}{8}=0.625$$ $$63\%=0.63$$

So the ascending order is:

$$\boxed{\frac{3}{5},0.62,\frac{5}{8},63\%}$$

Question 14

Convert $\frac{2}{5}$ into a decimal:

$$\frac{2}{5}=0.4$$

Now subtract:

$$1.15-0.4=0.75$$ $$\boxed{0.75}$$

As a fraction:

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

So $\frac{3}{4}$ is also correct.

Question 15

Find the equipment cost:

$$\frac{3}{8}\text{ of }640$$ $$640÷8=80$$ $$80\times 3=240$$

Find the transport cost:

$$25\%\text{ of }640=160$$

Total spent:

$$240+160=400$$

Money left:

$$640-400=240$$

Now find the percentage left:

$$\frac{240}{640}=\frac{3}{8}$$ $$\frac{3}{8}=0.375=37.5\%$$ $$\boxed{\pounds 240\text{ is left, which is }37.5\%\text{ of the original money.}}$$

Question 16

First convert the red part:

$$0.35=\frac{35}{100}=\frac{7}{20}$$

Use a common denominator of $60$:

$$\frac{5}{12}=\frac{25}{60}$$ $$\frac{7}{20}=\frac{21}{60}$$

The blue and red parts total:

$$\frac{25}{60}+\frac{21}{60}=\frac{46}{60}$$

The yellow part is:

$$\frac{60}{60}-\frac{46}{60}=\frac{14}{60}$$

Simplify:

$$\frac{14}{60}=\frac{7}{30}$$ $$\boxed{\frac{7}{30}}$$

Question 17

Find $10\%$:

$$780÷10=78$$

Then:

$$30\%=234$$ $$5\%=39$$

Therefore:

$$35\%=234+39=273$$ $$\boxed{273\text{ kg}}$$

Question 18

After a $20\%$ discount, Jack pays $80\%$ of the original price.

$$80\%=\pounds 132$$ $$10\%=\pounds 132÷8=\pounds 16.50$$ $$100\%=\pounds 16.50\times 10=\pounds 165$$ $$\boxed{\pounds 165}$$

A quick check: $20\%$ of £165 is £33, and:

$$\pounds 165-\pounds 33=\pounds 132$$

Question 19

Find the number of lollipops:

$$\frac{5}{9}\text{ of }3600$$ $$3600÷9=400$$ $$400\times 5=2000$$

Find the number of gummy bears:

$$20\%\text{ of }3600=720$$

Now subtract both from the total:

$$3600-2000-720=880$$ $$\boxed{880\text{ chocolate bars}}$$

Question 20

The £18 spent on pens is half of the money left after buying the pencil case.

So the amount left after the pencil case was:

$$\pounds 18\times 2=\pounds 36$$

That £36 is $\frac{2}{3}$ of Pippa’s starting money, because she spent $\frac{1}{3}$.

One third is:

$$\pounds 36÷2=\pounds 18$$

The whole amount is:

$$\pounds 18\times 3=\pounds 54$$ $$\boxed{\pounds 54}$$

Check:

$$\frac{1}{3}\text{ of }\pounds 54=\pounds 18$$ $$\pounds 54-\pounds 18=\pounds 36$$ $$\frac{1}{2}\text{ of }\pounds 36=\pounds 18$$

The check matches the information in the question.

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Key takeaways

You have completed questions 13–20 of the 50-question challenge. The most important habits from this lesson are:

• Fractions, decimals and percentages can represent the same value.
• Convert values into one form before comparing them.
• To find a fraction of an amount, divide by the denominator, then multiply by the numerator.
• To find a percentage, use efficient chunks such as $10\%$, $5\%$, and $1\%$.
• For a “what is left?” question, begin with the whole and subtract the known parts.
• For a reverse percentage question, identify what percentage the final amount represents before calculating.

Next, you will work on difficult ratio, proportion and scaling problems, where you will use similar ideas about parts, wholes, and multiplicative relationships.