1.3 Operations using numbers and calculator skills

How to use the memory keys on your calculator

The memory keys (M+, M\(-\), and MRC) allow you to do calculations in the calculator's memory (check what the equivalent keys are on your own calculator).

• The M+ key is used to add a number to the memory, or to add it to a number already in the memory.
• The M\(-\) key is used to subtract a number from the number in memory.
• If you press the MRC key once, the calculator displays the number stored in memory. If you press this key twice, the calculator's memory is cleared.
• When you use a memory key, the letter 'M' appears at the top of the display screen, showing that the number on the display has been stored in the calculator's memory.

This means that we can do longer calculations without having to write down the steps in between. It also gives you a way of doing calculations in the right order. Let's see how this works.

Compare the key sequence in the previous example to the key sequence: \(\text{200} + \text{2} \times \text{80} - \text{60}\). The memory key allows you to work without brackets.

Changing the sign of numbers

Practise changing the sign of a number on your calculator. For example, changing \(\text{10}\) to \(--\text{10}\). Some calculators have a [\(\pm\)] key, which changes the sign of a number.

If you see \(\pm\) (plus-minus) written before a number in other situations, it means the number is only approximate. It is only on the calculator that this key refers to a function that changes the sign of the number.

Brackets

We can use brackets to show the order in which operations happened in the situation. We want to show that Vaneshri should multiply the stall rental by \(\text{3}\) before subtracting it from the profit:

\(\text{40 000} - (\text{2 000} \times \text{3}) =\text{34 000}\)

BODMAS

If there are no brackets in a calculation, we use the BODMAS rule to remember in which order we must perform operations. This is a rule which says what the order must be:

If there are brackets involved then we must do the operation in the brackets first; then do the multiplication or division (it doesn't matter what the order is); and last, do the addition or subtraction (in any order).

Breaking down and multiplying

Breaking down numbers before multiplying is a useful trick for doing mental multiplication quickly. It is a rule that states that you can multiply a large number directly, or break it up into parts and multiply each part, and then add the answers together again. You will get the same answer either way.

For example, if you need to work out \(\text{4} \times \text{57}\) quickly:

\begin{align*} \text{4} \times \text{57}&= \text{4} \times (\text{50} + \text{7}) \text{ (Break up the number \text{57})}\\ &= (\text{4} \times \text{50}) + (\text{4} \times \text{7}) \text{ (Distribute over the parts of \text{57})}\\ &= \text{200} + \text{28}\\ &= \text{228} \end{align*}

It is easy to calculate \((\text{4} \times \text{50}) = \text{200}\) and \((\text{4} \times \text{7}) = \text{28}\) mentally, so you can do the whole calculation without writing it down.

When doing these calculations, look for ways to change the calculation so that you are multiplying by a multiple of ten, or using the multiplication tables that you can do mentally. Sometimes turning a number into a subtraction calculation helps.

Let's see how this rule works in the following examples.

Using grouping

Grouping numbers in different ways can also help us to multiply and add quickly.

There is a rule that states that in a calculation that only contains addition operations, we can group addition calculations in any order, and similarly, in a calculation that only contains multiplication operations, we can group the calculations in any order.

This helps us, because if we rearrange the numbers, we might find that we can do calculations by just looking at them!

In a calculation such as \(\text{25} \times \text{13} \times \text{4}\), it is easier to rewrite this as \(\text{25} \times \text{4} \times \text{13}\) because this becomes \((\text{25} \times \text{4}) \times 13 = \text{100} \times \text{13} = \text{1 300}\).

Multiplying by \(\text{10}\), \(\text{100}\) and \(\text{1 000}\) without a calculator

Another trick to keep in mind is how we multiply whole numbers by \(\text{10}\), \(\text{100}\) and \(\text{1 000}\).

• To multiply by \(\text{10}\), every digit moves to the left by one decimal place, and a zero moves into the units position.
• To multiply by \(\text{100}\), every digit moves to the left by two decimal places, and two zeros move into the tens and units positions.
• To multiply by \(\text{1 000}\), every digit moves to the left by three decimal places, and three zeros move into the hundreds, tens and units positions.

It is useful to do these calculations in a place value table, or in columns.

Adding and subtracting fractions

If you don't use a calculator, you first need to make sure that all fractions have the same denominator before you add or subtract them. Using a calculator often makes calculations with fractions much easier.

Two different methods for adding or subtracting fractions are given below. The method you choose depends on whether you want a common fraction or a decimal fraction as the answer.

Method 1:

To keep your answer as a common fraction, you need to change the denominators of all the fractions you are adding or subtracting to be the same.

• First multiply the denominators together to find a common denominator that will work for all of the fractions. Try to simplify it to make the calculations easier, but remember, it must still be a multiple of all of the original denominators. This demoninator is called the lowest common denominator (LCD).
• Next multiply the numerator of each fraction by the same number you multiplied with the denominator to get the LCD. (e.g. if the original denominator of a fraction is \(\text{3}\), and the LCD of the group of fractions is \(\text{6}\), we multiply the numerator by \(\text{2}\), because \(\text{3}\) \(\times\) \(\text{2}\) = \(\text{6}\)).

For example, how much is \(\frac{\text{1}}{\text{2}} + \frac{\text{1}}{\text{3}} + \frac{\text{1}}{\text{6}}\)?

\(\text{2} \times \text{3} = \text{6}\), so \(\text{6}\) is a common denominator for all three fractions.

Next we multiply each fraction's numerator by the \(\dfrac{\text{LCD}}{\text{denominator}}\).

\begin{align*} &\frac{\text{1}}{\text{2}} + \frac{\text{1}}{\text{3}} + \frac{\text{1}}{\text{6}}\\ & = \frac{(\text{1} \times \frac{\text{6}}{\text{2}}) + (\text{1} \times \frac{\text{6}}{\text{3}}) + (\frac{\text{6}}{\text{6}})}{\text{6}}\\ & = \frac{(\text{1} \times \text{3}) + (\text{1} \times \text{2}) + \text{1}}{\text{6}}\\ &= \frac{\text{6}}{\text{6}}\\ &= \text{1} \end{align*}

Method 2:

Treat each fraction like a separate division calculation, e.g. \(\frac{\text{4}}{\text{5}} = \text{4} \div \text{5}\).

Put the answers of the division calculations into the calculator's memory.

Add to or subtract from the memory and press MRC to find the answer.

Let's put this all together in some worked examples. These look quite complicated, but if you work through them step-by-step, you will see how they work!

What is a decimal fraction?

Think back to place value: Thousands; Hundreds; Tens and Units.

The bar below has been divided into ten equal parts; \(\text{1}\) is divided into \(\text{10}\). We say that \(\text{0,1}\) is one tenth. It is the only way in which calculators can write one tenth.

We have found that \(\text{1} \div \text{10} = \text{0,1}\).

Think back to fractions:

\begin{align*} \text{1} \div \text{2} &= \frac{\text{1}}{\text{2}}\\ \text{So } \text{1} \div \text{10} &= \frac{\text{1}}{\text{10}}\\ \text{1} \div \text{10} &= \frac{\text{1}}{\text{10}} = \text{0,1} \end{align*}

\(\text{0,1}\) is just another way of writing \(\frac{\text{1}}{\text{10}}\).

Converting between fractions and decimal fractions

To write a common fraction as a decimal fraction, we can use some different methods. First, we can convert the fraction to a fraction with a denominator of \(\text{10}\), \(\text{100}\) or \(\text{1 000}\). For example, \(\frac{\text{1}}{\text{4}} = \frac{\text{25}}{\text{100}} = \text{25}\%\).

How do we write fractions like thirds and sixths as decimal fractions? We can't write them with denominators of \(\text{10}\), \(\text{100}\) or \(\text{1 000}\). Instead, we use division.

Positive and negative numbers as opposites

Every positive number has an opposite negative number on the other side of zero. If you add these two numbers together, the answer is zero.

For example, \(\text{4} + (--\text{4}) = \text{0}\). This is the same as subtracting \(\text{4}\) from \(\text{4}\).