# Part 2: Decimals & Percentages | Beginners Guide to Year 7 Maths

In this article, we will go through a variety of decimal and percentage question types, provide some worked examples and give you a list of checkpoint questions to test your knowledge!

Starting Year 7 Maths may sound a little daunting. But don’t fear! With this article, we’ll help you ease into Year 7 decimals and percentages with clear explanations and examples followed by some checkpoint questions to see how you’ve progressed.

## NSW Syllabus Outcomes

 NSW Syllabus Outcomes Syllabus Outcomes Explanation Multiply and divide decimals using efficient written strategies This means that we will show you how to multiply or divide $$0.0786$$ by $$10^n$$ Round decimals to a specified number of decimal places This means that you will know how to round numbers like $$4.5675$$ to $$2$$ decimal places $$4.57$$ Investigate terminating and recurring decimals This means that you know how to deal with numbers like $$1.353535353$$ and $$1.\dot{35}$$ in questions. Connect fractions, decimals and percentages and carry out simple conversions This means that you can convert a decimal $$(0.5)$$ into a percentage $$(0.5\%)$$ or fraction $$(\frac{1}{2})$$ and vice versa. Find percentages of quantities and express one quantity as a percentage of another This means that you know how to express $$500mm$$ as a percentage of $$10m$$ . Solve problems involving the use of percentages, including percentage increases and decreases This means that you know how to problem solve questions involving percentages $$(eg. 56\%)$$.

## Assumed knowledge:

Students should be familiar with the terminology used when describing fractions. They should also be familiar with simple arithmetic involving fractions such as expressing a mixed number as an improper fraction.

## Decimals

### Converting decimals to fractions

A decimal is an alternative method of writing a fraction where the denominator is a power of ten.

For example,

\begin{align*}
\frac{1}{10} = 0.1; \frac{1}{100} = 0.01; \frac{1}{1000} = 0.001 \ \text{etc}
\end{align*}

When converting a decimal to a mixed fraction:

1. Keep anything prior to the decimal point to remain as your whole number in your mixed fraction.
2. For the fraction component, write out the digits after the decimal point on the numerator. On the denominator, write $$1$$ followed by as many zeroes as there are digits after the decimal place.
3. Finally, simplify the fraction where possible

Example:

1. Express the decimal $$32.4605$$ as a fraction, simplifying where possible.

Solution:

For this question, first note that anything before the decimal place will be kept for the whole number in the mixed fraction.

In this question, this number is $$32$$.

For the fraction component, write out the digits after the decimal point $$(4605)$$ on the numerator of your fraction.

Also note that there are $$4$$ digits after the decimal place, meaning that your denominator will be $$10000$$ ($$4$$ zeroes).

This allows you to obtain the mixed fraction:

\begin{align*}
32 \frac{4605}{10000}
\end{align*}

Finally, simplify this fraction:

\begin{align*}
32\frac{921}{2000}
\end{align*}

Note that this process is equivalent to breaking down the decimal into fractions considering each decimal place separately and adding them:

\begin{align*}
32.4605 = 32 + \frac{4}{10} + \frac{6}{100} + \frac{0}{1000} + \frac{5}{10000} = 31\frac{921}{2000}
\end{align*}

### Converting fractions to decimals

Recall that a fraction is another way of representing a division.

Hence, by performing a long division and dividing the numerator by the denominator, we can express a fraction as a decimal!

Example

1. Express $$8\frac{7}{8}$$ as a decimal.

Solution:

To do this example, all we need to do is convert the fraction part of the mixed fraction into a decimal.

This can be done using long division:

Hence $$8 \frac{7}{8}$$

Note that another way of converting fractions to decimals is converting their denominators to powers of ten:

e.g. $$\frac{17}{25} = \frac{17 \times 4}{25 \times 4} = \frac{68}{100} = 0.68$$

### Terminating and recurring decimals

Terminating decimals are decimals that end after a certain number of decimal places.

Recurring decimals have one or more digits that repeat indefinitely

e.g. $$0.44444444, 0.53535353$$ etc.

To express a recurring decimal, dots are placed over the repeating digits.

• If one digit is recurring, a dot is placed over the repeating digit

$$0.55555555555… = 0.\dot{5}$$

• If two digits are recurring, dots are placed over the two repeating digits

$$0.12463636363…= 0.124\dot{6}\dot{3}$$

• If three or more digits are recurring, dots are placed over the first and last recurring digit

$$0.14721472147214….= 0.\dot{1}47\dot{2}$$

### Ordering decimal

To order decimals, simply write all the given decimals to the same number of decimal places.

(Note: adding zeroes to the end of a decimal does NOT change its value!)

Once this is done, you can simply compare the size of the digits.

Example

1. Write $$0.47, 0.0047, 0.047, 0.407$$ in ascending order

Solution:

To do this example, write the given decimals to $$4$$ decimal places:

$$0.4700; 0.0047; 0.0470; 0.4070$$

Then compare the size of the digits and write the decimals from smallest to largest (ascending order)

$$0.0047 < 0.0470 < 0.4070 < 0.4700$$

### Multiplication of decimals

When multiplying a decimal by $$10^n$$ (a power of $$10$$), move the decimal point $$n$$ places to the right.

e.g. Here, $$10^4$$ is the same as $$10 000$$, so move the decimal point $$4$$ places to the right.

$$0.0000426 \times 10 000 = 0.426$$

• When multiplying a decimal by a multiple of $$10$$, complete the following process:

For example, $$25.146 \times 2000$$

Step 1: Express the multiple of $$10$$ in terms of its factors:

$$25.146 \times 2000 = 25.146 \times 2 \times 10^3$$

Step 2: Multiply the numbers (keeping the position of the decimal point):

$$25.146 \times 2 \times 10^3 = 50.292 \times 10^3$$

Step 3: As discussed earlier, move the decimal point the appropriate number of places to the right (in this case 3)

$$50.292 \times 10^3=50 292$$

• When multiplying a decimal by another decimal, complete the following process:

For example, $$47.236 \times 0.028$$

Step 1: Ignoring the decimal points, multiply the 2 numbers

$$47236 \times28=1322608$$

Step 2: Determine the total number of digits after the decimal points for both numbers.

For this question, there are 6 digits after the decimal point ($$3$$ from the first number, and $$3$$ from the second number).

The number that you obtain here is the number of digits after the decimal point in your final answer.

Step 3: For this question, we move the decimal point so that there are $$6$$ digits after it.

$$47.236 \times 0.028=1.322608$$

### Scientific notation

Scientific Notation (standard notation) is a convenient method of writing extremely large and small numbers in a compact form.

It is written as a product of a number between $$1$$ and $$10$$ (can be a decimal), and a power of $$10$$.

To write a number in scientific notation, move the decimal point such that it is placed after the first non-zero digit.

Here, the power of $$10$$ is given by the number of places the decimal point needs to be moved to the left or right.

• Suppose we are asked to represent the large number $$73269.8$$ in scientific notation.

Moving the decimal point to the first non-zero digit, we obtain the number $$7.32698$$ (this will be the number between $$1$$ and $$10$$ used in our product).

To get to this decimal, we had to move the decimal point $$4$$ places to the left (this makes $$4$$ the power of $$10$$ in our notation).

Here, note the positive power of $$10$$ for the large number.

Hence, in scientific notation:

$$73269.8 = 7.32698×10^4$$

• Alternatively, suppose we are asked to represent the small number $$0.000004036$$ in scientific notation.

Moving the decimal point to the first non-zero digit, we obtain the number $$4.036$$ (this will be the number between $$1$$ and $$10$$ used in our product).

To get to this decimal, we had to move the decimal point to the right by $$6$$ places (this makes $$-6$$ the power of $$10$$ in our notation).

Here, note the negative power of $$10$$ for the small number.

Hence, in scientific notation:

$$0.000004036 = 4.036 \times 10^{(-6)}$$

### Division of decimals

Similar to the multiplication of decimals, dividing a decimal by $$10n$$ (a power of $$10$$) involves moving the decimal point $$n$$ places to the left.

Eg. $$425.67 \div 100$$

Here, $$100$$ is the same as $$102$$.

So, move the decimal point $$2$$ places to the left to obtain:

$$425.67 \div 100=4.2567$$

• To divide a decimal by multiples of $$10$$, complete the following process:

Step 1: Express the multiple of $$10$$ in terms of its factors. Make sure to remember the brackets (order of operations)

$$47.24 \div 5000= 47.24 \div (5×10^3)$$

Step 2: Divide the numbers (keeping the position of the decimal point)

$$47.24 \div 5 \div 10^3 = 9.448 \div 10^3$$

Step 3: As discussed earlier, move the decimal point the appropriate number of places to the left (in this case 3)

$$9.448 \div 10^3=0.009448$$

• To divide a decimal by another decimal, complete the process shown in this example:

1. $$0.00336 \div 0.042$$

Solution:

Move the decimal point to the right in both numbers such that the divisor is a whole number.

For this question, move the decimal point 3 places to the right:

$$0.00336 \div 0.042 = 3.36 \div 42$$

Then complete the division (using long division):

$$3.36 \div 42 = 0.08$$

### Rounding

Sometimes, it is useful to have a shorter representation of decimal numbers.

Rounding refers to reducing the digits of a number so that they are approximately equal.

When we express a number in terms of $$‘n’$$ decimal places, it means that we must have only $$‘n’$$ digits after the decimal point.

If you are rounding decimals to a specific number of decimal places:

• If the next digit is $$0, 1, 2, 3, 4$$ round down.
• If the next digit is $$5, 6, 7, 8, 9$$ round up

Example:

1. Approximate $$4.28$$ to $$1$$ decimal place

Solution:

To round to $$1$$ decimal place, we need to look at the second decimal place value to determine if we have to round up or down.

In this case, as the second decimal place is $$8$$, we must round up:

Hence $$4.28 \approx 4.3$$ (correct to $$1$$ d.p.)

Note that $$\approx$$ means “is approximately equal to” and “d.p.” means “decimal places

## Percentages

A percentage is a number that represents a fraction of $$100$$ (denoted by the percentage sign $$\%$$).

Percentages, fractions and decimals are all interchangeable.

### Converting percentage to a fraction

To convert a percentage to a fraction, substitute the symbol with a fraction of denominator $$100$$.

e.g. $$15 \% = \frac{15}{100}$$

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

Note: Don’t forget to simplify after doing this process!

### Converting fractions to percentages

To convert any fraction to a percentage, multiply by $$100\%$$ then simplify.

e.g. $$\frac{7}{15} = \frac{7}{15} \times 100 \% = 46 \frac{2}{3} \%$$

### Converting percentages to decimals and vice versa

To convert from a percentage to a decimal, take the value of the percentage and move the decimal place $$2$$ places the left.

e.g. $$124 \% = 1.24$$

To convert from a decimal to a percentage, do the opposite and move the decimal point $$2$$ places the right.

Note: If there is a fraction in the percentage, convert the mixed fraction into a decimal as per the method discussed earlier!

## Application of Percentages

### Percentage composition

Often times, a question will ask you to express one quantity as a percentage of another.

In order to do this, write a fraction with the ‘part amount’ in the numerator and the ‘whole amount’ in the denominator.

Then, convert this to a percentage as per the methods discussed above:

$$\frac{Part Amount}{Whole Amount} \times 100 \%$$

Note: Make sure your quantities have the SAME UNITS or else this formula won’t work!

Example:

1. Express the quantity $$430 mm$$ as a percentage of $$40 m$$.

Solution:

First, we must convert the quantities into the same units.

Here, $$40$$ metres is equivalent to $$40000$$ millimetres.

Then, use the formula to express as a percentage:

$$\frac{430}{40 000} \times 100 \% = 1/075 \%$$

### Percentage of a quantity

Another application of percentages applied in the real world is the process of finding a certain percentage of a given quantity

e.g. discounts in shopping, tax rates, depreciation of property etc.

To achieve this, simply convert your percentage to a fraction as per the methods discussed above.

Then, multiply this fraction by the quantity (you can consider the quantity as a fraction with denominator)

Example:

1. Edmund pays $$32 \%$$ of his wage in tax and earns $$146$$ per day. How much does he pay in tax?

Solution:

First, convert $$32 \%$$ into a fraction as per the methods discussed above:

$$32 \% = \frac{32}{100} = \frac{8}{25}$$

Then, multiply the fraction by the total quantity:

$$\frac{8}{25} \times \frac{146}{1} = \frac{1168}{25} = 46.72$$

### Percentage increase/decrease

Percentages are often used to measure an increase or decrease in a quantity.

To determine the ending quantity following an increase/decrease by a given percentage, complete the following procedure:

1. Add/Subtract the percentage increase/decrease to/from $$100 \%$$
e.g. Increase of $$8 \%$$ is equivalent to $$108\%$$ while a decrease by $$8 \%$$ is equivalent to $$92 \%$$
2. Find this percentage of the initial quantity

Example:

1. Cameron aims to increase his sleep by $$20\%$$. He currently sleeps for $$6$$ hours. What amount of sleep does he hope to reach:

First, express the increase as a percentage:

$$100 \% + 20 \% = 120 \%$$

Then, find the percentage of the initial quantity as per methods discussed above:

$$120 \%$$ of $$6$$ hours $$= \frac{120}{100} \times 6 = \frac{710}{100} = \frac{36}{5} = 7 \frac{1}{5}$$ hours of sleep

Note: An alternative method to do this question is to directly measure the percentage increase/decrease in terms of quantity and add/subtract it from the original quantity.

For the example above, the working out would be as follows:

Amount increased $$= 20 \%$$ of $$6$$ hours $$= \frac{20}{100} \times = 1 \frac{1}{5}$$

Ending quantity $$=$$ Original Quantity $$+$$ Amount Increased $$= 6 + 1 \frac{1}{5} = 7 \frac{1}{5}$$

### Percentage change

Following a change in the value of a quantity, there are many applications where it is useful to express this change as a percentage of the original size or value.

When comparing two quantities (with the same units!), you can determine the percentage increase or decrease by:

Percentage increase $$= \frac{Increase}{Original Value} \times 100 \%$$

Percentage decrease $$= \frac{Decrease}{Original Value} \times 100\%$$

Example:

1. The price of a calculator decreased from $$\50$$ to $$\20$$. What is the percentage decrease?

Solution:

From this question, it is quite easy to see that the calculator has decreased by $$\30$$ in value $$(\50 – \20)$$

Hence, the percentage decrease can be determined using the formula:

$$\frac{\30}{\50} \times 100 \% = 60\%$$

### Percentages and the unitary method

The unitary method (as discussed in the Fractions Blog Post) can also be applied to percentages if a percentage is given and a total amount is required.

Here, we must:

1. Find $$1%$$ (one unit) of the quantity
2. Multiply by $$100$$ to obtain the value for the full quantity ($$100\%$$)

Example:

1. James is a maths lecturer and received $$\477$$ after a bonus of $$6\%$$ was added to his original pay. What was his pay before the bonus?

Solution:

\begin{align*}
New Pay &= 100\% + 5\%\\
105\% &= \477 \\ \end{align*} Find $$1\%$$ of the quantity: \begin{align*} 1 \% = \frac{\ 477}{106}
\end{align*}

Multiply by $$100$$ to find original pay:

\begin{align*}
100 \% = 100 \times \frac{$477}{106} = \$ 450
\end{align*}

## Summary

### Decimals:

1. Converting a decimal to a fraction involves writing out the digits after the decimal point on the numerator, and on the denominator, writing $$1$$ followed by as many zeroes as there are digits after the decimal place.
2. Converting a fraction to a decimal involves performing long division or changing the denominator to a power of  $$10$$
3. Multiplying a decimal by $$10n$$ causes the decimal points to move $$n$$ places to the right. Dividing a decimal by $$10n$$ causes the decimal points to move $$n$$ places to the left.
4. To multiply a decimal by a decimal, ignore the decimal points and multiply the numbers, then place the decimal point so that your multiplied number has the same number of digits after the decimal point as in the question
5. To divide a decimal by a decimal, change the divisor into a whole number by moving the decimal point right in both numbers. Then divide as per usual.
6. When rounding decimals to a specific number of decimal places, if the next digit is $$0, 1, 2, 3, or 4$$ round down. If the next digit is $$5, 6, 7, 8, or 9$$ round up.

### Percentages:

1. To convert from a percentage to a fraction, substitute the symbol with a fraction of denominator $$100$$.
2. To convert a fraction to a percentage, multiply the fraction by $$100\%$$.
3. To convert from a percentage to a decimal, move the decimal point  $$2$$ places to the left.
4. To convert a decimal to a percentage, move the decimal point  $$2$$ places to the right.
5. Percentage composition of a quantity as part of a whole:
$$\frac{Part Amount}{Whole Amount} \times 100 \%$$
6. Percentage change of a quantity:

$$\% Increase = \frac{Increase}{Original Value} \times 100 \%$$

$$\% Decrease = \frac{Decrease}{Original Value} \times 100 \%$$

## Checkpoint Questions

1. Convert $$47 \frac{13}{25}$$ into a decimal.

2. Express $$4\frac{11}{15}$$ as a recurring decimal

3.Dallas sells $$4300$$ computer parts at $$\72.95$$ each. He makes a profit of $$\15.37$$ on each computer part he sells.

1. What is his total selling price?
2. What is his total profit?

4. Find $$3.76 \times 4.23$$ and express this decimal as a fraction

5. Arrange the following in ascending order: $$\frac{7}{15}; 45\%; 0.457$$

6. Udayveer was given $$306$$ marks in a test that was weighted out of $$450$$ marks. Express Udayveer’s mark as a percentage of total marks in the test.

7. Jess made a chocolate cake that was made up of $$400$$ grams of chocolate, $$50$$ grams of berries, $$350$$ grams of fudge and $$100$$ grams of nuts. What percentage of the chocolate cake is:

1. Chocolate?
2. Berries?
3. Fudge?
4. Nuts?

8. Aidan left a glass of water out in the sun on a hot day. $$35 \%$$ of the water evaporated out after $$1$$ hour. If the glass of water originally had $$675 mL$$, how much water is left after $$1 hour$$?

9. Jason has a rectangular paddock which is $$50 m$$ long and $$25 m$$ wide. He plans on increasing the lengths of his paddock by $$30\%$$ and the widths by [/latex]20\%[/latex]. What is the percentage increase in the area of his paddock?

10. Isaac owns a bookstore that makes a profit of $$25\%$$ when they sell a maths textbook for $$\480$$. How much should he sell the maths textbook if he wants to make a $$45\%$$ profit?

## Solutions

1. $$47.52$$

2. $$5.7\dot{3}$$

3.

1. $$4300 \times \72/95 = \ 313 685$$
2. $$4300 \times \15.37 = \66 091$$

4. $$15.9048 = 15 \frac{1131}{1250}$$

5. $$45\%: 0.457; \frac{7}{15}$$

6. $$68\%$$

7.

1. $$\frac{400}{900} = 44.\dot{4}\%$$
2. $$\frac{50}{900} = 5.\dot{5}\%$$
3. $$\frac{350}{900} = 38.\dot{8}\%$$
4. $$\frac{100}{900} = 11.\dot{1}\%$$

8. $$\frac{65}{100} \times 675ml = 438.75 ml$$

9. $$56\%$$

10. $$\556.80$$

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