To estimate is to try to get close to an answer without actually doing the required calculations with the given
numbers. An estimate may also be called an approximation. The difference between an estimate and the actual
answer is called the error.

Calculating with “easy” numbers that are close to given numbers is a good way to obtain approximate
answers (or estimates), for example:

To approximate \(764 + 829\), you could calculate \(800 + 800\) to get the approximate answer \(1\ 600\),
with an error of \(7\).

To approximate \(84 \times 178\), you could calculate \(80 \times 200\) to get the approximate answer \(16\
000\), with an error of \(1\ 048\).

Worked Example 1.6: Finding approximate answers

Calculate with “easy” numbers close to the given numbers to produce approximate answers for
this product. Do not use a calculator. When you have made your approximations, use a calculator to find the
exact answer and calculate the error.

\[78 \times 46\]

Find easy numbers close to the given numbers.

\[80 \times 40\]

You could also take \(80\) and \(50\) as your easy numbers.

Multiply without using a calculator.

\[80 \times 40 = 3\ 200\]

Use a calculator to find the actual answer.

\[78 \times 46 = 3\ 588\]

Find the error in calculation.

The actual answer is bigger than the approximation.

\[3\ 588 - 3\ 200 = 388\]

The error is \(388\), so the estimation was an underestimation.

Worked Example 1.7: Finding approximate answers

Calculate using easy numbers, and without using a calculator. When you have made your approximations, use a
calculator to find the precise answer and calculate the error.

\[78 \times 178\]

Find easy numbers close to the given numbers.

\[80 \times 200\]

Multiply.

\[80 \times 200 = 16\ 000\]

Use a calculator to find the actual answer.

\[78 \times 178 = 13\ 884\]

Find the error in calculation.

The actual answer is smaller than the approximation.

\[16\ 000 - 13\ 884 = 2\ 116\]

The error is \(2\ 116\), so the estimation was an overestimation.

This is quite a large error. Imagine that you were cutting wood to make furniture and your error was
\(2\ 116 \text{ mm}\). If each piece of wood was \(2\ 116 \text{ mm}\) wider than it should be, you
would have a problem making your furniture!

Another two easy numbers you could have chosen are \(70 \times 200 = 14\ 000\). The second number is
quite far from \(200\), so it is better to choose a lower first number. The error in this calculation is
then much lower: \(14\ 000 - 13\ 884 = 116\).

Exercise 1.1: Approximate the answers to large
products

Calculate with easy numbers close to the given numbers to produce approximate answers for each product
below.
Do not use a calculator. When you have made your approximations, use a calculator to find the precise
answers.

\[88 \times 56\]

\[77 \times 98\]

\[94 \times 275\]

\[71 \times 198\]

\[1\ 250 \times 79\]

\[88 \times 56\]

Approximation: \(90 \times 50 = 4\ 500\)

Actual answer: \(4\ 928\)

\[77 \times 98\]

Approximation: \(80 \times 100 = 8\ 000\)

Actual answer: \(7\ 546\)

\[94 \times 275\]

Approximation: \(90 \times 300 = 27\ 000\)

Actual answer: \(25\ 850\)

\[71 \times 198\]

Approximation: \(70 \times 200 = 14\ 000\)

Actual answer: \(14\ 058\)

\[1\ 250 \times 79\]

Approximation: \(1\ 000 \times 80 = 80\ 000\)

Actual answer: \(98\ 750\)

Methods for adding, subtracting and multiplying

Let’s revise some basic addition, subtraction and multiplication facts.

Remember that the sum is what you get after you have done your addition. We can add in any order.

\[44 + 41 = 85\]

The sum of \(44\) and \(41\) is \(85\).

The difference is what is left after you have done your subtraction. We must subtract in the given
order.

\[55 - 15 = 40\]

The difference between \(55\) and \(15\) is \(40\).

There are lots of ways to subtract numbers from each other. This regrouping method is just one
example.

\[(55 - 5) - 10 = 50 - 10 = 40\]

The product is what you get after you have done your multiplication. We can multiply in any order.

\[5 \times 10 = 50\]

The product of \(5\) and \(10\) is \(50\).

The quotient is what is left after you have done your division. We must divide in the given order.

To determine the quotient when \(30\) is divided by \(5\), we must divide \(30\) by \(5\).

\[30 \div 5 = 6\]

The quotient is \(6\).

temp text

Methods for addition

Numbers can be added by thinking of their parts as we say the numbers. For example, we say \(4\ 994\) as
‘four thousand nine hundred and ninety-four’. This can be written in expanded notation as \(4\ 000 +
900 + 90 + 4\).

Similarly, we can think of \(31\ 837\) as \(30\ 000 + 10\ 000 + 800 + 30 + 7\).

\(31\ 837 + 4\ 994\) can be calculated by working with the various kinds of parts separately. To make this
easy, the numbers can be written below each other so that the units are below the units, the tens below the tens
and so on, as shown here.

To achieve this, the first step is to add the digits in the ones column: \(7 + 4 = 11\). Only write the digit
“\(1\)” of the \(11\). The \(10\) of the \(11\) is remembered and added to the \(3\) in the tens
column.

We say the \(10\) is carried from the ones column to the tens column. The same is done when the tens parts are
added to get \(13\): only the digit “\(3\)” is written (in the tens column, so it means \(30\)), and
the \(1\) is carried to the next column. This \(1\) is actually \(100\), so it needs to be carried to the
hundreds column.

Impilo Enterprises plans a new computerised training facility in their existing building. The training
manager has to keep the total expenditure budget under \(\text{R } 1 \text{ million}\). This is what she has
written so far.

Architects and builders

\(\text{R} 102\ 700\)

Painting and carpeting

\(\text{R} 42\ 600\)

Security doors and blinds

\(\text{R} 52\ 000\)

Data projector

\(\text{R } 4\ 800\)

\(25\) new secretary chairs

\(\text{R} 50\ 400\)

\(24\) desks for work stations

\(\text{R} 123\ 000\)

\(1\) desk for presenter

\(\text{R} 28\ 000\)

\(25\) new computers

\(\text{R} 300\ 000\)

\(12\) colour laser printers

\(\text{R} 38\ 980\)

Work out the total cost of all the items the training manager has budgeted for.

Break each number into parts (expand the numbers).

Architects and builders

\(\text{R} 102\ 700\)

\(102\ 700 = 100\ 000 + 2\ 000 + 700\)

Painting and carpeting

\(\text{R} 42\ 600\)

\(42\ 600 = 40\ 000 + 2\ 000 + 600\)

Security doors and blinds

\(\text{R} 52\ 000\)

\(52\ 000 = 50\ 000 + 2\ 000\)

Data projector

\(\text{R} 4\ 800\)

\(4\ 800 = 4\ 000 + 800\)

\(25\) new secretary chairs

\(\text{R} 50\ 400\)

\(50\ 400 = 50\ 000 + 400\)

\(24\) desks for work stations

\(\text{R} 123\ 000\)

\(123\ 000 = 100\ 000 + 20\ 000 + 3\ 000\)

\(1\) desk for presenter

\(\text{R} 28\ 000\)

\(28\ 000 = 20\ 000 + 8\ 000\)

\(25\) new computers

\(\text{R} 300\ 000\)

\(300\ 000 = 300\ 000\)

\(12\) colour laser printers

\(\text{R} 38\ 980\)

\(38\ 980 = 30\ 000 + 8\ 000 + 900 + 80\)

Write the numbers below each
other and add up the numbers in columns.

Be careful as you place the numbers according to their place value!

There are many ways to find the difference between two numbers. For example, to find the difference between
\(267\) and \(859\), you can think of the numbers as they may be written on a number line.

We may think of the distance between \(267\) and \(859\) as three steps: from \(267\) to \(300\), from \(300\)
to \(800\), and from \(800\) to \(859\). How big are each of these three steps?

This number line shows that \(859 - 267\) is \(33 + 500 + 59\). This means that \(859 - 267 = 592\).

Like addition, subtraction can also be done by working with the different parts in which we say (or expand) the
numbers. For example, \(8\ 764 − 2\ 352\) can be calculated as follows:

\(8\) thousand − \(2\) thousand = \(6\) thousand

\(7\) hundred − \(3\) hundred = \(4\) hundred

\(6\) tens − \(5\) tens = \(1\) ten

\(4\) units − \(2\) units = \(2\) units

So, \(8\ 764 − 2\ 352 = 6\ 412\).

Subtraction by parts is more difficult in some cases, for example \(6\ 213 − 2\ 758\):

\(6\ 000 - 2\ 000 = 4\ 000\) This step is easy, but the following steps cause problems:

\(200 - 700 = \ ?\)

\(10 - 50 = \ ?\)

\(3 - 8 = \ ?\)

One way to overcome these problems is to work with negative numbers:

\(200 - 700 = - 500\)

\(10 - 50 = - 40\)

\(3 - 8 = - 5\)

Now, we add positive and negative answers to get the result:

In this method, it is easy to make mistakes if you are not careful with the minus sign. Fortunately, the parts
and sequence of work may be rearranged to overcome these problems, as shown below. We regroup both numbers to
make subtraction in columns easier:

First estimate the answers to the nearest \(100\ 000\) or \(10\ 000\) or \(1\ 000\). Then calculate.

\[238\ 769 - 141\ 453\]

\[856\ 333 - 739\ 878\]

\[65\ 244 - 39\ 427\]

\[238\ 769 - 141\ 453\]

Estimate: \(240\ 000 − 140\ 000 = 100\ 000\)

Actual: \(97\ 316\)

\[856\ 333 - 739\ 878\]

Estimate: \(860\ 000 − 740\ 000 = 120\ 000\)

Actual: \(116\ 455\)

\[65\ 244 - 39\ 427\]

Estimate: \(65\ 000 − 40\ 000 = 25\ 000\)

Actual: \(25\ 817\)

A method of multiplication

Just as with addition and subtraction, we can use the method of breaking up numbers into parts. For example, to
find the product \(7 \times 4\ 598\), we can expand \(4\ 596\) like this:

\(7 \times 4\ 000 = 28\ 000\)

\(7 \times 500 = 3\ 500\)

\(7 \times 90 = 630\)

\(7 \times 8 = 56\)

The four partial products can now be added to get the answer, which is \(32\ 186\). It is convenient to write
the work in vertical columns for units, tens, hundreds and so on, as shown below.

The answer can be produced with less writing by “carrying” parts of the partial answers to the next
column, when working from right to left in the columns.

Only the 6 of the product \(7 \times 8\) is written down instead of \(56\). The \(50\) is kept in mind, and
added to the \(630\) obtained when \(7 \times 90\) is calculated in the next step.

Worked Example 1.13: Multiplying whole numbers

Calculate each of the following. Do not use a calculator.

\[27 \times 649\]

Write the two numbers in columns, start
with the bigger number.

Use your calculator to check that this answer is correct!

Exercise 1.3

Calculate each of the following. Do not use a calculator now.

\[29 \times 549\]

\[78 \times 1\ 750\]

\[641 \times 99\]

Use your calculator to check your answers for Question 1. Redo the questions for which you had the
wrong
answers.

\[29 \times 549 = 15\ 921\]

\[78 \times 1\ 750 = 136\ 500\]

\[641 \times 99 = 63\ 459\]

Calculate each of the following. Do not use a calculator now.

\[97 \times 176\]

\[74 \times 197\]

\[97 \times 176 = 17\ 072\]

\[74 \times 197 = 14\ 578\]

Long division method

Long division is a useful method for dividing large numbers without a calculator. Essentially, you break the
division problem into lots of similar steps, and follow a sequence.

Worked Example 1.15: Dividing whole numbers

Use long division method to find the answer to \(9\ 396 \div 27\).

Set up the calculation in columns.

Divide.

Ask the question: “How many times does \(27\) go into \(93\)?”

You choose \(93\) because \(27\) cannot go into \(9\), the very first digit.

The answer is “\(3\) times”, because \(27 \times 3 = 81\). You write \(3\) above the division
line and \(81\) below \(93\).

Subtract and bring the next digit down.

\[93 - 81 = 12\]

Divide again (repeat Step 2).

Ask the question: “How many times does \(27\) go into \(129\)?”

The answer is “\(4\) times”, because \(27 \times 4 = 108\). You write \(4\) above the division
line and \(108\) below \(129\).

Subtract and bring the next digit down (repeat
Step 3).

Divide again (repeat Step 2).

Ask the question: “How many times does \(27\) go into \(216\)?”

The answer is “\(8\) times”, because \(27 \times 8 = 216\). You write \(8\) above the division
line and \(216\) below \(216\).

Subtract.

We now stop this calculation. The remainder of the division is \(0\).

The answer is \(9\ 396 \div 27 = 348\).

Worked Example 1.16: Dividing whole numbers

Use long division to calculate \(4\ 125 \div 12\).

Set up the calculation in columns.

Divide.

Ask the question: “How many times does \(12\) go into \(41\)?”

You choose \(41\) because \(12\) cannot go into \(4\), the very first digit.

The answer is “\(3\) times”, because \(12 \times 3 = 36\). You write \(3\) above the division
line and \(36\) below \(41\).

Subtract and bring the next digit down.

\[41 - 36 = 5\]

Divide again (repeat Step 2).

Ask the question: “How many times does \(12\) go into \(52\)?”

The answer is “\(4\) times”, because \(12 \times 4 = 48\). You write \(4\) above the division
line and \(48\) below \(52\).

Subtract and bring the next digit down (repeat
Step 3).

Divide again (repeat Step 2).

Ask the question: “How many times does \(12\) go into \(45\)?”

The answer is “\(3\) times”, because \(12 \times 3 = 36\). You write \(3\) above the division
line and \(36\) below \(45\).

Subtract.

We now stop this calculation. The remainder of the division is \(9\).

So, the answer is \(4\ 125 \div 12 = 343\) with a remainder of \(9\).

Brenda bought \(64\) goats, all at the same price. She paid \(\text{R} 5\ 440\) in total. What was the
price for each goat?

Your first step can be to work out how much Brenda would have paid if she paid \(\text{R} 10\) per
goat, but you can start with a bigger step if you wish.

\[5\ 440 \div 64 = \text{R}85\]

Mosibudi has \(\text{R} 2\ 850\) and he wants to buy candles for his sister’s wedding reception.
The candles cost \(\text{R} 48\) each. How many candles can he buy?

\(2\ 850 \div 48 = 59 \text{ rem } 18\), so he can buy \(59\) candles.

Calculate each of the following, without using a calculator.