Using a Calculator

How to key negative numbers, powers, roots, π, fractions and whole expressions into a scientific calculator, how to round what the display shows, and how an estimate catches a wrong key.

Before you start

The order of operations: brackets first; then powers and roots; then multiplication and division, from left to right; then addition and subtraction, from left to right. So $20 - 12 \div 4 = 20 - 3 = 17$ .

$(-3)^{2}$ and $-3^{2}$ are different. In $(-3)^{2}$ the brackets show that $-3$ is squared: $(-3) \times (-3) = 9$ . In $-3^{2}$ only the 3 is squared, and the negative sign is put in front afterwards: $-(3 \times 3) = -9$ .

The Keys of a Scientific Calculator

A scientific calculator works out a whole expression in the correct order of operations, provided it is entered correctly. Find each of the keys below on yours: the labels differ a little from one model to another.

Key	What it does
$(-)$	The negative key. It gives the number after it a negative sign.
$-$	The subtraction key. It goes between two numbers. It is not the same key as $(-)$ .
$($ $)$	Brackets. What is inside them is worked out first.
$\frac{\square}{\square}$	The fraction key, for entering a fraction.
$\text{S}\Leftrightarrow\text{D}$	Changes an answer between a fraction and a decimal. (On some calculators it is marked $\text{F}\Leftrightarrow\text{D}$ or CHANGE.)
$x^{2}$ $x^{3}$ $x^{\square}$	Square, cube, and any other power.
$\sqrt{\square}$ $\sqrt[3]{\square}$	Square root and cube root.
$\pi$	The value of $\pi$ , to more digits than the display shows. Use this key for $\pi$ unless a question gives a value to use.
$\rightarrow$	The right arrow of the cursor pad, the large key with four arrows near the display. It moves out of a box on the display.

Some of these are printed above a key instead of on it. To use one of those, press SHIFT (or 2ndF) first.

Check your own calculator

If the fraction, root or power key opens an empty box on the display, type inside the box, then press $\rightarrow$ to leave it before you go on. For a fraction, fill the top box, move down, then fill the bottom box. If no box opens, type the numerator, then the fraction key, then the denominator. If the root key writes $\surd($ with an open bracket instead of a box, type the number and then close the bracket with $)$ before you go on.
To enter a mixed number such as $2\frac{1}{3}$ , use the mixed-number key if your calculator has one (on many it is SHIFT and then the fraction key), or enter $(2 + \frac{1}{3})$ with brackets. An answer may appear as an improper fraction: $-\frac{23}{6}$ is $-3\frac{5}{6}$ .
If an answer appears as a fraction, or with a root sign or $\pi$ in it, press $\text{S}\Leftrightarrow\text{D}$ to see it as a decimal. Many calculators show $1.75$ as $\frac{7}{4}$ first. So wherever this lesson says that the display shows a decimal and yours shows a fraction, a root sign or $\pi$ , press $\text{S}\Leftrightarrow\text{D}$ .
In the rows of keys below, $\rightarrow$ and $\text{S}\Leftrightarrow\text{D}$ are drawn with a dashed edge. Press them only if your calculator needs them: $\rightarrow$ to leave a box, and $\text{S}\Leftrightarrow\text{D}$ to change the answer to a decimal. If your root key writes $\surd($ with an open bracket, press $)$ where the dashed $\rightarrow$ is drawn.

Example 1: Negative numbers, powers and roots

Use a calculator to evaluate
(a) $5 - (-8)$
(b) $(-3)^{2}$
(c) $-3^{2}$
(d) $(-2)^{5}$
(e) $\sqrt[3]{15}$
(f) $2\pi + \sqrt{5}$

Solution

(a) $5 - (-8)$

Keys: 5 $-$ $(-)$ 8 $=$ Display: 13

(b) $(-3)^{2}$

Keys: $($ $(-)$ 3 $)$ $x^{2}$ $=$ Display: 9

Keys: $(-)$ 3 $x^{2}$ $=$ Display: −9

(d) $(-2)^{5}$

Keys: $($ $(-)$ 2 $)$ $x^{\square}$ 5 $=$ Display: −32

(e) $\sqrt[3]{15}$

Keys: $\sqrt[3]{\square}$ 15 $=$ Display: 2.466212074

(f) $2\pi + \sqrt{5}$

Keys: 2 $\times$ $\pi$ $+$ $\sqrt{\square}$ 5 $=$ $\text{S}\Leftrightarrow\text{D}$ Display: 8.519253285

In (a), $-$ subtracts and $(-)$ gives 8 its negative sign.

In (b), the brackets make the calculator square $-3$ : $(-3) \times (-3) = 9$ . In (c) there are no brackets, so it squares 3 first and then makes the result negative: $-(3 \times 3) = -9$ . The calculator follows the order of operations, just as you do on paper.

(e) and (f) do not come out exactly. $\sqrt[3]{15}$ , $\pi$ and $\sqrt{5}$ are irrational numbers: their digits never end and never settle into a block that repeats. A ten-digit display shows each answer rounded to ten digits.

$(-3)^{2} = 9$ but $-3^{2} = -9$ . Put a negative number in brackets before you raise it to a power.

Giving an Answer to a Number of Decimal Places

The display often shows more digits than an answer needs. A question says how many decimal places to give, and the display is rounded to that many.

Rounding the display

Find the last decimal place that you are asked to keep, and look at the digit after it.
If that digit is 5 or more, round up: add 1 to the last digit you keep. If it is 4 or less, leave the last digit as it is.
Write every decimal place that was asked for, even when the last one is 0, and say how the answer was rounded: "(correct to 2 decimal places)".

Rounding to a number of significant figures comes in the next chapter.

Example 2: Rounding the display

A calculator shows $\sqrt[3]{15}$ as 2.466212074 and $\sqrt{63.9}$ as 7.993747557.
(a) Write $\sqrt[3]{15}$ correct to 1 decimal place, to 2 decimal places and to 3 decimal places.
(b) Write $\sqrt{63.9}$ correct to 2 decimal places and to 1 decimal place.

Solution

(a) To 1 decimal place: the digit after the first decimal place is 6, so round up to $2.5$ . To 2 decimal places: the digit after the second decimal place is 6, so round up to $2.47$ . To 3 decimal places: the digit after the third decimal place is 2, so the 6 stays, and the answer is $2.466$ .

(b) To 2 decimal places: the digit after the second decimal place is 3, so the 9 stays, and the answer is $7.99$ .

To 1 decimal place: the digit after the first decimal place is 9, so round up. Adding 1 to the last digit of $7.9$ gives $8.0$ . Keep the 0: it shows that the answer is correct to 1 decimal place.

(a) $\sqrt[3]{15} = 2.5$ (correct to 1 decimal place), $2.47$ (correct to 2 decimal places), $2.466$ (correct to 3 decimal places)
(b) $\sqrt{63.9} = 7.99$ (correct to 2 decimal places), $8.0$ (correct to 1 decimal place)

Fractions and Mixed Numbers

With the fraction key, the calculator works in fractions and gives the answer as a fraction, which is exact. This is a good way to check a fraction sum that you have worked on paper.

Example 3: Fractions

Use a calculator to evaluate $-\frac{5}{8} + \frac{1}{4}$ . Give the answer as a fraction and as a decimal.

Solution

Press $(-)$ , enter $\frac{5}{8}$ with the fraction key, press $+$ , enter $\frac{1}{4}$ , then press $=$ . The display shows $-\frac{3}{8}$ .

Press $\text{S}\Leftrightarrow\text{D}$ . The display changes to $-0.375$ . Press it again to go back to the fraction.

Check on paper: $-\frac{5}{8} + \frac{2}{8} = -\frac{3}{8}$ .

$-\frac{5}{8} + \frac{1}{4} = -\frac{3}{8} = -0.375$

Example 4: Mixed numbers

Use a calculator to evaluate $2\frac{1}{3} - 6\frac{1}{6}$ . Give the answer as a mixed number, and as a decimal correct to 2 decimal places.

Solution

Estimate: about $2 - 6 = -4$ .

Enter each mixed number with the mixed-number key, or enter $(2 + \frac{1}{3}) - (6 + \frac{1}{6})$ with both pairs of brackets. Then press $=$ .

The display shows $-3\frac{5}{6}$ or the improper fraction $-\frac{23}{6}$ . Since $23 \div 6 = 3$ remainder 5, $-\frac{23}{6} = -3\frac{5}{6}$ .

Press $\text{S}\Leftrightarrow\text{D}$ for the decimal. The display shows $-3.833333333$ . The 3s go on without end, so the display has cut the decimal short: the fraction is exact and the display is not. The digit after the second decimal place is 3, so the answer is $-3.83$ .

Check on paper: $\frac{7}{3} - \frac{37}{6} = \frac{14}{6} - \frac{37}{6} = -\frac{23}{6}$ .

Without the second pair of brackets, the calculator would add $\frac{1}{6}$ instead of subtracting it, and give $-3\frac{1}{2}$ . That is close to the estimate, but wrong: an estimate does not catch every slip, so check the brackets as well.

$2\frac{1}{3} - 6\frac{1}{6} = -3\frac{5}{6} = -3.83$ (correct to 2 decimal places)

A Whole Expression in One Go

A long calculation is entered as one expression, so that the calculator keeps every digit of every part. Two things keep the answer right: brackets in the correct places, and an estimate made before you press a key.

Method: a whole expression in one go

Estimate the answer first, using easy numbers.
Enter the whole expression. A fraction line is a division: put brackets round the whole numerator and round the whole denominator.
Do not round part-way. Copy more digits than you need from the display, then round once, to the number of decimal places asked for.
Compare the answer with your estimate. If they are far apart, a key was wrong: enter the expression again.

Example 5: Brackets round the numerator and the denominator

Evaluate $\dfrac{2.4^{2} - \sqrt{7.3}}{1.6 \times 0.35}$ , giving your answer correct to 3 decimal places.

Solution

Estimate: $2.4^{2}$ is about 6 and $\sqrt{7.3}$ is a little less than 3, so the numerator is about 3. The denominator is about $1.5 \times 0.4 = 0.6$ . The answer should be near $3 \div 0.6 = 5$ .

Enter $(2.4^{2} - \sqrt{7.3}) \div (1.6 \times 0.35)$ in one go.

Keys: $($ 2.4 $x^{2}$ $-$ $\sqrt{\square}$ 7.3 $\rightarrow$ $)$ $\div$ $($ 1.6 $\times$ 0.35 $)$ $=$ $\text{S}\Leftrightarrow\text{D}$

The display shows $5.460979969$ . The fourth decimal place is 9, so round up: $5.461$ . This is close to the estimate.

Without the brackets round the denominator, the calculator divides by 1.6 and then multiplies by 0.35. It shows $0.668\ldots$ , which the estimate tells you is wrong.

$\dfrac{2.4^{2} - \sqrt{7.3}}{1.6 \times 0.35} = 5.461$ (correct to 3 decimal places)

Wrong Keys, and How an Estimate Catches Them

A calculator works out exactly what is keyed, right or wrong. These are the four slips that happen most often, each with the wrong display it gives. An estimate made beforehand shows up the first three at once.

Slip 1: The subtraction key for a negative sign

Wanted: $-4.8 \times 2.5$

Estimate: A negative number times a positive number is negative: about $-5 \times 2.5 = -12.5$ .

Suppose the answer to the calculation before this one was 13, and it is still on the display.

Wrong keys: $-$ 4.8 $\times$ 2.5 $=$ Wrong display: 1

Many calculators take a calculation that starts with the subtraction key to mean "the last answer minus …": they work out $13 - 4.8 \times 2.5$ and show 1. Other calculators show an error message instead. If the display has been cleared first, some calculators take the key as a negative sign and show $-12$ . Do not rely on that: use $(-)$ for a negative sign every time.

Right keys: $(-)$ 4.8 $\times$ 2.5 $=$ Display: −12

The estimate is negative and the wrong display is positive, so a key was wrong. Using $(-)$ where the subtraction key is meant is just as wrong: keying 5, then $(-)$ , then 3 for $5 - 3$ gives an error message or a wrong answer.

Slip 2: No brackets round the numerator or the denominator

Wanted: $\dfrac{18.6 + 7.4}{2.5 \times 1.3}$

Estimate: The numerator is 26. The denominator is about 3, because $2.5 \times 1.3$ is a little more than 2.5. So the answer is near $26 \div 3$ , which is about 9.

Wrong keys: 18.6 $+$ 7.4 $\div$ 2.5 $\times$ 1.3 $=$ $\text{S}\Leftrightarrow\text{D}$ Wrong display: 22.448

The calculator follows the order of operations: it divides only 7.4 by 2.5, multiplies the result by 1.3, and adds 18.6 last. With brackets round the numerator but none round the denominator, it divides 26 by 2.5 and then multiplies by 1.3, and shows 13.52.

Right keys: $($ 18.6 $+$ 7.4 $)$ $\div$ $($ 2.5 $\times$ 1.3 $)$ $=$ Display: 8

Neither 22.448 nor 13.52 is near 9, so the estimate rules both out. The right display, 8, is near it.

Slip 3: A negative number squared without brackets

Wanted: $(-1.5)^{2} + 4$

Estimate: The square of a number is never negative, so the answer cannot be less than 4.

Wrong keys: $(-)$ 1.5 $x^{2}$ $+$ 4 $=$ $\text{S}\Leftrightarrow\text{D}$ Wrong display: 1.75

Without brackets, the calculator squares 1.5 first and then makes the result negative: $-2.25 + 4 = 1.75$ .

Right keys: $($ $(-)$ 1.5 $)$ $x^{2}$ $+$ 4 $=$ $\text{S}\Leftrightarrow\text{D}$ Display: 6.25

1.75 is less than 4, so it cannot be right.

Slip 4: Rounding part-way

Wanted: $2.4 \div 0.7 \times 35$

Estimate: $2.4 \div 0.7$ is a little more than 3, and $3 \times 35 = 105$ , so expect an answer somewhat more than 105.

Wrong keys: 2.4 $\div$ 0.7 $=$ $\text{S}\Leftrightarrow\text{D}$ Wrong display: 3.428571429

This was written down as 3.43 and keyed in again.

Wrong keys: 3.43 $\times$ 35 $=$ $\text{S}\Leftrightarrow\text{D}$ Wrong display: 120.05

The division was rounded before the multiplication, and the small error in 3.43 was then multiplied by 35.

Right keys: 2.4 $\div$ 0.7 $\times$ 35 $=$ Display: 120

An estimate cannot catch this slip, because the wrong answer is close to the right one. Only the habit prevents it: key the whole expression, or go straight on from the answer on the display, and round once at the end.

When to Use a Calculator, and When Not To

A calculator is for arithmetic that is long, or that cannot be finished by hand: decimals with many digits, roots that do not come out exactly, and $\pi$ . It does not choose the calculation for you, and it cannot tell whether you keyed what you meant.

Leave the calculator alone when

the sum is quick in your head, such as $-7 + 12$ or $25 \times (-4)$ . Keying it takes longer and adds the risk of a wrong key.
the answer is a simple fraction. On paper, $\frac{2}{3} + \frac{1}{6} = \frac{5}{6}$ exactly. As a decimal, $\frac{5}{6}$ recurs, so no display can show all of it.
the root is of a perfect square or a perfect cube: $\sqrt{144} = 12$ and $\sqrt[3]{27} = 3$ .
the question says "without a calculator" or asks for an exact value.

When you do use a calculator, write down the expression that you are working out before you write the answer. Then you, or your teacher, can see what was keyed.

Example 6: Calculator or not?

For each calculation, decide whether a calculator is needed. Then find the answer.
(a) $-8 \times 7 + 6$
(b) $\frac{5}{6} - \frac{1}{3}$
(c) $17.38 \times 4.6 - 52.9 \div 2.3$
(d) $\sqrt{196} \div (-7)$
(e) the exact value of $5 \div 3$

Solution

(a) No calculator: $-8 \times 7 = -56$ , and $-56 + 6 = -50$ .

(b) No calculator: $\frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$ , which is exact.

(c) Calculator. Estimate first: $17 \times 5 = 85$ and $50 \div 2 = 25$ , so the answer is about $85 - 25 = 60$ . Enter the expression as it is written: the calculator multiplies and divides before it subtracts. The display shows $56.948$ , which is close to the estimate.

(d) No calculator: $14 \times 14 = 196$ , so $\sqrt{196} = 14$ , and $14 \div (-7) = -2$ .

(e) No calculator: $5 \div 3 = \frac{5}{3} = 1\frac{2}{3}$ . Keyed as a division, the display shows $\frac{5}{3}$ or $1.666666667$ . In the decimal the 6s go on without end and the display has rounded the last digit, so $1.666666667$ is not the exact value.

(a) $-50$
(b) $\frac{1}{2}$
(c) $56.948$
(d) $-2$
(e) $1\frac{2}{3}$

Common Mistakes

Pressing $-$ for a negative sign. $-$ subtracts. The sign of a negative number is the $(-)$ key.
Entering $-3^{2}$ for $(-3)^{2}$ . A negative number that is raised to a power needs brackets.
Leaving out the brackets round a numerator or a denominator. A fraction line is a division of the whole numerator by the whole denominator.
Rounding part-way, or keying $3.14$ when the $\pi$ key is meant.
Cutting the display off instead of rounding it. Correct to 2 decimal places, 2.466212074 is $2.47$ , not $2.46$ .
Dropping a final 0. Correct to 1 decimal place, 7.993747557 is $8.0$ , not 8.
Copying the display without an estimate. The calculator cannot tell that a key was wrong. The estimate can.

Practice

Work each question before you open its solution. Estimate before you key. Question 6 is to be done without a calculator. Where a solution says that the display shows a decimal and yours shows a fraction, a root sign or $\pi$ , press $\text{S}\Leftrightarrow\text{D}$ .

Question 1

Use a calculator to evaluate
(a) $(-2.5)^{2}$
(b) $-2.5^{2}$
(c) $(-1.2)^{3}$
(d) $-7 - (-4.6)$
(e) $(-2)^{6}$
Explain why the answers to (a) and (b) are different.

ShowHide solution

Use $(-)$ for every negative sign, and $-$ only for the subtraction in (d).

In (a) the brackets make the calculator square $-2.5$ : $(-2.5) \times (-2.5) = 6.25$ . In (b) it squares 2.5 first and then makes the result negative.

(c) Use the cube key, $x^{3}$ , or the power key, $x^{\square}$ , with 3. Three negative factors give a negative product: $(-1.2) \times (-1.2) \times (-1.2) = -1.728$

(d) $-7 - (-4.6) = -7 + 4.6 = -2.4$

(e) Put $-2$ in brackets, then use the power key, $x^{\square}$ , with 6. Six negative factors, an even number, give a positive product: $2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$

(a) $6.25$
(b) $-6.25$
(c) $-1.728$
(d) $-2.4$
(e) $64$

Question 2

Use a calculator to evaluate each of the following, correct to 2 decimal places.
(a) $\sqrt{20}$
(b) $\sqrt[3]{75}$
(c) $5\pi - \sqrt{3}$
(d) $(-2.6)^{4}$
(e) $\sqrt{5^{2} + 7^{2}}$

ShowHide solution

(a) Estimate: $4^{2} = 16$ and $5^{2} = 25$ . Since $16 < 20 < 25$ , $\sqrt{20}$ is between $\sqrt{16} = 4$ and $\sqrt{25} = 5$ . The display shows $4.472135955$ . The third decimal place is 2, so round down.

(b) Estimate: $4^{3} = 64$ and $5^{3} = 125$ , so $\sqrt[3]{75}$ is between 4 and 5. The display shows $4.217163327$ . The third decimal place is 7, so round up.

(c) Estimate: about $5 \times 3 - 2 = 13$ . Enter $5 \times \pi - \sqrt{3}$ with the $\pi$ key, and press $\text{S}\Leftrightarrow\text{D}$ if the answer appears with $\pi$ or a root sign in it. The display shows $13.97591246$ . The third decimal place is 5, so round up.

(d) Put $-2.6$ in brackets, then use the power key, $x^{\square}$ , with 4. Four negative factors, an even number, give a positive answer. The display shows $45.6976$ . The third decimal place is 7, so round up: adding 1 to the last digit of $45.69$ gives $45.70$ . Keep the 0.

(e) Everything under the root sign must go inside the root: type $5^{2} + 7^{2}$ in the box, or between $\surd($ and $)$ . This is $\sqrt{74}$ , which is between 8 and 9. The display shows $8.602325267$ . The third decimal place is 2, so round down, and keep the 0 in the second decimal place. If the root covers only $5^{2}$ , the display shows 54.

(a) $4.47$
(b) $4.22$
(c) $13.98$
(d) $45.70$
(e) $8.60$
Each answer is correct to 2 decimal places.

Question 3

A calculator shows $\sqrt[3]{200}$ as 5.848035476. Write $\sqrt[3]{200}$ correct to
(a) 1 decimal place,
(b) 2 decimal places,
(c) 4 decimal places.
(d) Jun Wei writes 5.84 for part (b). What did he do wrong?
(e) Priya writes 5.848 for part (c). What is wrong with her answer?

ShowHide solution

(a) The digit after the first decimal place is 4, so the 8 stays.

(b) The digit after the second decimal place is 8, so round up: the 4 becomes 5.

(d) He cut the display off after two decimal places instead of rounding it.

(e) 5.848 has only three decimal places. The 0 in the fourth decimal place must be written, to show that the answer is correct to 4 decimal places.

(a) $5.8$
(b) $5.85$
(c) $5.8480$
(d) He did not round up.
(e) The final 0 is missing.

Question 4

Use a calculator to evaluate $-\frac{5}{6} \div \frac{10}{9} + \frac{1}{2}$ . Give the answer as a fraction and as a decimal.

ShowHide solution

Press $(-)$ , enter $\frac{5}{6}$ , press $\div$ , enter $\frac{10}{9}$ , press $+$ , enter $\frac{1}{2}$ , then press $=$ . The display shows $-\frac{1}{4}$ . Press $\text{S}\Leftrightarrow\text{D}$ for the decimal.

Check on paper: $-\frac{5}{6} \times \frac{9}{10} = -\frac{3}{4}$ , and $-\frac{3}{4} + \frac{1}{2} = -\frac{1}{4}$ .

$-\frac{5}{6} \div \frac{10}{9} + \frac{1}{2} = -\frac{1}{4} = -0.25$

Question 5

Use a calculator to evaluate $1\frac{3}{4} - 4\frac{2}{5}$ . Give the answer as a mixed number and as a decimal.

ShowHide solution

Estimate: about $2 - 4.5 = -2.5$ .

Enter each mixed number with the mixed-number key, or enter $(1 + \frac{3}{4}) - (4 + \frac{2}{5})$ with both pairs of brackets. Without the second pair, the calculator would add $\frac{2}{5}$ instead of subtracting it.

The display shows $-2\frac{13}{20}$ or the improper fraction $-\frac{53}{20}$ . Since $53 \div 20 = 2$ remainder 13, $-\frac{53}{20} = -2\frac{13}{20}$ .

Press $\text{S}\Leftrightarrow\text{D}$ for the decimal: $-2.65$ .

Check on paper: $\frac{7}{4} - \frac{22}{5} = \frac{35}{20} - \frac{88}{20} = -\frac{53}{20}$ .

$1\frac{3}{4} - 4\frac{2}{5} = -2\frac{13}{20} = -2.65$

Question 6

Do not use a calculator for this question.
(a) Evaluate $-15 + 40 - 25$ .
(b) Evaluate $\frac{3}{4} - \frac{1}{8}$ .
(c) Evaluate $\sqrt[3]{125} \times (-6)$ .
(d) Siti keys $7 \div 3$ into her calculator and writes down $2.333333333$ as the exact answer. Is she right? Give a reason.

ShowHide solution

(a) $-15 + 40 = 25$ , and $25 - 25 = 0$ . This is quicker in your head than on the keys.

(b) $\frac{6}{8} - \frac{1}{8} = \frac{5}{8}$ , which is exact.

(d) $7 \div 3 = 2\frac{1}{3} = 2.333\ldots$ , in which the 3s go on without end. The display stops after ten digits, so what it shows is not the exact value. The fraction is.

(a) $0$
(b) $\frac{5}{8}$
(c) $-30$
(d) No. The exact value is $2\frac{1}{3}$ .

Question 7

Use a calculator to evaluate $\dfrac{\sqrt[3]{20} + 4.7}{2.5^{2} - 3.8}$ , giving your answer correct to 2 decimal places.

ShowHide solution

Estimate: $\sqrt[3]{20}$ is a little less than $\sqrt[3]{27} = 3$ , so the numerator is about 7.5. The denominator is $6.25 - 3.8$ , which is about 2.5. The answer should be near $7.5 \div 2.5 = 3$ .

Enter $(\sqrt[3]{20} + 4.7) \div (2.5^{2} - 3.8)$ with both pairs of brackets. The display shows $3.026292905$ .

The third decimal place is 6, so round up.

$\dfrac{\sqrt[3]{20} + 4.7}{2.5^{2} - 3.8} = 3.03$ (correct to 2 decimal places)

Question 8

Three pupils use their calculators to evaluate $\dfrac{38.4 - 9.6}{1.6 \times 4.5}$ . Their displays show 81, 11.4 and 4.
(a) Estimate the answer.
(b) Which display is correct?
(c) What did each of the other two pupils leave out?

ShowHide solution

(a) The numerator is about $38 - 10 = 28$ and the denominator is about $1.5 \times 5 = 7.5$ . $28 \div 7.5$ is a little less than 4.

(b) Only 4 is near the estimate. Check: $(38.4 - 9.6) \div (1.6 \times 4.5) = 28.8 \div 7.2 = 4$ .

(c) 81 comes from leaving out the brackets round the denominator: the calculator works out $28.8 \div 1.6 = 18$ and then multiplies by 4.5.

11.4 comes from using no brackets at all: the calculator works out $9.6 \div 1.6 \times 4.5 = 27$ first, and then $38.4 - 27 = 11.4$ .

(a) About 4
(b) The display 4 is correct.
(c) The pupil with 81 left out the brackets round the denominator. The pupil with 11.4 left out both pairs of brackets.

Question 9

Use a calculator to arrange $\sqrt{7}$ , $-\sqrt{3}$ , $2.6$ , $-1\frac{3}{4}$ and $\sqrt[3]{30}$ in ascending order.

ShowHide solution

$-1\frac{3}{4}$ and $-\sqrt{3}$ are both between $-2$ and $-1$ , and $2.6$ and $\sqrt{7}$ are both between 2 and 3, so here the decimals are needed.

For $\sqrt{7}$ the display shows $2.645751311$ , for $\sqrt{3}$ it shows $1.732050808$ and for $\sqrt[3]{30}$ it shows $3.107232506$ . Also $-1\frac{3}{4} = -1.75$ .

Of the two negative numbers, $-1.75$ is further from 0 than $-1.732\ldots$ , so $-1\frac{3}{4}$ is the smaller. $2.6$ is less than $2.645\ldots$ , and $\sqrt[3]{30}$ is the only number greater than 3.

In ascending order: $-1\frac{3}{4}$ , $-\sqrt{3}$ , $2.6$ , $\sqrt{7}$ , $\sqrt[3]{30}$

Question 10

At a ski resort, the temperature at 6 a.m. on five mornings was $-3.6$ °C, $1.8$ °C, $-7.5$ °C, $-0.9$ °C and $2.4$ °C. Use a calculator to find
(a) the difference between the highest and the lowest of these temperatures,
(b) the average of the five temperatures.

ShowHide solution

(a) The highest is $2.4$ and the lowest is $-7.5$ . Difference $= 2.4 - (-7.5)$ . Use $-$ for the subtraction and $(-)$ for the negative sign of 7.5. The display shows $9.9$ .

(b) Average $= \dfrac{-3.6 + 1.8 + (-7.5) + (-0.9) + 2.4}{5}$ . Estimate: $-4 + 2 + (-8) + (-1) + 2 = -9$ , and $-9 \div 5$ is about $-2$ .

Enter $(-3.6 + 1.8 + (-7.5) + (-0.9) + 2.4) \div 5$ , with brackets round the whole sum and the $(-)$ key for each negative sign. The display shows $-1.56$ , which is close to the estimate.

Without the brackets round the sum, the calculator divides only 2.4 by 5. It shows $-9.72$ , which the estimate tells you is wrong.

(a) The difference is 9.9 °C.
(b) The average temperature is $-1.56$ °C.

Question 11

A bicycle wheel has a diameter of 0.7 m. How far does the bicycle travel when the wheel makes 120 complete turns? Use the $\pi$ key of your calculator, and give your answer correct to 1 decimal place.

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In one turn the bicycle travels the circumference of the wheel, which is $\pi \times 0.7$ m.

Distance $= 120 \times \pi \times 0.7$ . Estimate: $120 \times 3 \times 0.7 = 252$ .

Enter $120 \times \pi \times 0.7$ with the $\pi$ key. The display shows $263.8937829$ . The second decimal place is 9, so round up.

With 3.14 in place of the $\pi$ key, the answer would be 263.8, which is wrong in the last digit. Use the $\pi$ key unless a question gives a value for $\pi$ .

The bicycle travels 263.9 m (correct to 1 decimal place).

Question 12

A square garden has an area of 150 m². Fencing costs $12.50 per metre. Use a calculator to find
(a) the length of a side of the garden, correct to 2 decimal places,
(b) the cost of fencing all four sides, correct to the nearest cent.

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(a) Side $= \sqrt{150}$ m. Since $12^{2} = 144$ , expect a little more than 12. The display shows $12.24744871$ . The third decimal place is 7, so round up.

(b) Cost in dollars $= 4 \times \sqrt{150} \times 12.5$ . Enter it in one go. The display shows $612.3724357$ .

Using the rounded 12.25 m gives $4 \times 12.25 \times 12.5 = 612.50$ , which is 13 cents too much. Round only at the end.

(a) A side of the garden is 12.25 m long (correct to 2 decimal places).
(b) The fencing costs $612.37.

Key Points

On a calculator, $(-)$ gives a number its negative sign and $-$ subtracts.
$(-3)^{2} = 9$ but $-3^{2} = -9$ : put a negative number in brackets before you raise it to a power.
The fraction key gives an exact answer. A mixed number is entered with its own key, or as a whole number plus a fraction, in brackets.
Use the $\pi$ key for $\pi$ unless a question gives a value to use.
Estimate first. Enter the whole expression, with brackets round the numerator and round the denominator, and compare the display with your estimate.
Round once, at the end, to the number of decimal places asked for. Round up when the next digit is 5 or more, and keep a final 0.
A display is often rounded, so it is not always exact. Work in your head or in fractions when that is quicker or when an exact value is wanted.