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London School of Hygiene & Tropical Medicine
BASIC MATHS SUPPORT SESSIONS
Session 1  Basic Arithmetic Notes
1. Fractions
A fraction is a portion of a whole e.g. (one half), (one quarter), 1/3 (one third), 5/8 (five eighths).
The top number is the numerator and the bottom number is the denominator.
Fractions can be cancelled down if we can divide both the numerator and the denominator by the same number, e.g. 4/6 ( 2/3. We can expand fractions by multiplying the numerator and the denominator by the same number, e.g. 1/4 ( 3/12.
A mixed fraction is a whole number and a fraction, e.g. 1 (one and a half). We can also write this as a fraction 3/2.
To add and subtract fractions
We need to rewrite the fractions so that they have the same denominator; we call this the common denominator.
e.g. 1/4 + 2/7
We can find the common denominator by multiplying together the individual denominators, in this example 4 7 = 28. Remember that if we change the bottom number we must also change the top number, so 1/4 becomes 7/28 and 2/7 becomes 8/28.
Another way is to find the smallest number that can be divided by both the denominators.
e.g. 5/8  1/40 = 25/40  1/40 = 24/40. (40 is the smallest number which can be divided by both 8 and 40).
We can often cancel down the answer e.g. 24/40 = 12/20 = 3/5.
Check if we can cancel the fractions before doing the calculation, if we can then it often makes the numbers we are working with easier to handle.
e.g. 4/40 + 35/70 = 1/10 + 1/2 = 1/10 + 5/10 = 6/10 = 3/5 .
If we have mixed fractions then one method for dealing with them is to turn them into improper (top heavy) fractions first,
e.g. 1  3/4 = 3/2 3/4 = 6/4 3/4 = 3/4.
To multiply fractions
Multiply the top numbers together and multiply the bottom numbers together,
e.g. 2/3 4/7 = 24/37 = 8/21
1 3/4 = 3/2 3/4 = 9/8 = 11/8
To divide fractions
When we divide one fraction by another, we multiply the first by the reciprocal of the second. We obtain the reciprocal by inverting (turning upside down) the fraction.
e.g. 1/2 2/3 = 1/2 3/2 = 3/4
3/8 11/2 = 3/8 3/2 = 3/8 2/3 = 6/24 = 1/4.
2. Decimals
Decimals are fractions where the denominators are 10 and multiples of 10 such as 100, 1000 etc.
e.g. 0.3 = 3/10, 0.03 = 3/100, 0.003 = 3/1000, 2.3 = 2 3/10, 23.003 = 23 3/1000
We convert a fraction such as 3/8 to a decimal by dividing 3 by 8 to obtain 0.375
We convert a decimal to a fraction by taking the number before the decimal point as the whole number and the numbers after the decimal point as the fraction. One decimal place gives us a fraction out of 10, 2 decimal places gives a fraction out of 100, 3 decimal places gives a fraction out of 1000 etc.
To multiply a decimal by 10 we move the decimal point one place to the right.
To multiply by 100, move the decimal point two places to the right etc.
e.g. 45.67 10 = 456.7 0.04 100 = 4
To divide a decimal by 10 we move the decimal point one place to the left.
To divide by 100, move the decimal point two places to the left etc. Note that we put in zeroes as place markers to ensure that the decimal point is in the right place.
e.g. 45.67 10 = 4.567 0.04 100 = 0.0004
3. Percentages
Percentages are fractions out of 100.
e.g. 7% means 7 out of 100 = 7/100, 35% = 35/100, 40% = 40/100 = 2/5, 140% = 140/100 = 7/5
They can also be written as decimals.
e.g. 7% ( 0.07, 35% ( 0.35, 130% ( 1.30
To change a fraction to a percentage we multiply the fraction by 100 and cancel down.
e.g. 4/5 ( 4/5 100 = 400/5 = 80%
To find a percentage of a number we take the percentage as a fraction and multiply by the number.
e.g. 20% of 80 = 20/100 80 = 2/10 80 = 160/10 =16
120% of 80 = 120/100 80 = 12/10 80 = 96
To increase/decrease a number by a percentage
Suppose we want to increase 45 by 40%
First find 40% of 45 and then add that to 45.
40% of 45 equals 40/100 45 = 4/10 45 = 180/10 =18.
Then we add 18 to 45 to give 63.
If we were decreasing 45 by 40% we would subtract 18 from 45 to give 27.
Another way to increase 45 by 40% is to find 140% of 45
= 140/100 45 = 14/10 45 = 630/10 = 63.
Similarly to decrease 45 by 40% take 60% of 45
= 60/100 45 = 6/10 45 = 3/5 45 = 135/5 = 27.
4. Rounding
Rounding whole numbers
We can round whole numbers to the nearest ten, the nearest hundred, the nearest thousand etc.
e.g. Round 79 to the nearest 10.
7 9
tens units
To round this to the nearest ten, take the number at the tens column which is 7, and look at the number in the next column. If the number in the next column is 5 or bigger then we round up. Here the number in the next column is 9 so we round up and the answer is 80.
(Remember when rounding we need the zero to keep the right place value).
e.g. Round 72 to the nearest 10.
7 2
tens units
To round this to the nearest ten, take the number at the tens column which is 7, and look at the number in the next column. If the number in the next column is 5 or bigger then we round up. Here the number in the next column is 2 so we dont round up, the 7 stays the same and the answer is 70.
e.g. 712719 is seven hundred and twelve thousand, seven hundred and nineteen
7 1 2 7 1 9
hundred thousand ten thousand thousand hundreds tens units
To round this to the nearest thousand, take the number at the thousand column which is 2, and look at the number in the next column. If the number in the next column is 5 or bigger then we round up. Here the number in the next column is 7 so we round up and the answer is 713000. Notice that we must set the zeroes in as place markers for the last three columns, since we are rounding to the nearest thousand.
Rounding decimals
We can round decimals in the same way.
e.g. Round 2.718 to 1 decimal place and to 2 decimal places.
2 . 7 1 8
units tenths hundredths thousandths
units 1 decimal place 2 decimal places 3 decimal places
To round this to 1 decimal place, take the number at the 1 decimal place column which is 7, and look at the number in the next column. If the number in the next column is 5 or bigger then we round up. Here the number in the next column is 1 so we dont round up, the 7 stays the same and the answer is 2.7.
To round this to 2 decimal places, take the number at the 2 decimal place column which is 1, and look at the number in the next column. If the number in the next column is 5 or bigger then we round up. Here the number in the next column is 8 so we round up, the 1 becomes a 2 and the answer is 2.72.
5. Ratio and proportion
Ratio is a comparison between similar quantities. We can write a ratio as e.g. 3 to 2 or 3:2 or as a fraction 3/2.
We write ratios using whole numbers and we write a ratio in its simplest terms
e.g. 6:4 = 3:2,
12:4 = 3:1,
1:0.25 = 1: = 4:1
Proportions
e.g. Divide 40 in the ratio 5:3.
Total number of parts is 5 + 3 = 8.
Amount of each part is 40 8 = 5
Amount of first part is 55 = 25
Amount of second part is 35 = 15
So 40 divided in the ratio 5:3 is 25:15.
Using ratio
e.g. if two lengths are in the ratio 5:3 and the shorter length is 12 what is the longer length?
We have 5:3 = ?:12 i.e. 5/3 = ?/12
We need to multiply the denominator, 3, by 4 to get 12. We then multiply the numerator, 5, by 4 to get 20 which is the answer.
6. Order of operations and calculator use
Order of operations
Let us calculate 24 6 + 2.
We can work it out this way : 24 6 + 2 = 24 (6 + 2) = 24 8 = 3.
Alternatively we can calculate: 24 6 + 2 = (24 6) + 2 = 4 + 2 = 6.
Since we obtain two completely different results depending on which order we do the calculation, we can see that the order is vitally important for obtaining the required answer. Therefore, we need a set of rules to determine in which order to do the arithmetic operations.
When doing a calculation the order of operations is as follows:
1) Brackets
2) Exponents (see Session 3)
3) Multiplication and Division
4) Addition and Subtraction
In the above example, the correct order in which to do the calculation, as it is given, is:
24 6+2 = (24 6) + 2 = 4 + 2 = 6.
Note that we do the calculation from left to right obeying the order of operations along the way.
e.g. (2 3) + (127) = 6 + 5 = 11
24 4 + 6 2 = 6 + 12 = 18
If there are brackets within brackets, we start with the innermost bracket first.
e.g. 5+(10(186)) = 5 + (103) = 5 + 30 = 35
Using a calculator
To turn on the CASIO fx85GT PLUS calculator, press the blue button in the top righthand corner marked ON.
To turn off the CASIO fx85GT PLUS calculator, press SHIFT (the blue button in the top lefthand corner) followed by the AC key (the right most key of the two orange buttons).
For basic arithmetic we need to use the +, , , , = keys. These keys can all be found in the bottom righthand corner of the CASIO fx85GT PLUS calculator. The number keys are in the bottom lefthand corner. The ( and ) keys are in the middle of the calculator.
For simple calculations using CASIO fx85GT PLUS, press MODE 1 to choose COMP from the menu list, then press SHIFT MODE 2 to choose Line10 (to make numbers display with decimals). The MODE key is next to the ON button.
To clear the screen, use the AC button. Use the orange DEL button to remove a mistaken entry.
Use the SHIFT function key if function required is in brown/green/yellow above the key. When doing more complex calculations, it is likely that you will need to use the memory function to hold the intermediate value of the calculation.
As a simple example, on CASIO fx85GT PLUS, to use memory to calculate
EMBED Equation.3
key in:
to store this value (1591) in memory (i.e. using the STO function).
Then key in:
Your calculator should then report the answer as 1708.
The RCL key is 4 keys underneath the SHIFT key, while the M+ key is 4 keys underneath the ON key.
To clear this number (1708) from your calculators memory simply store the number zero in the memory by keying in:
The CASIO fx85GT PLUS calculator will also remember the result from the last time you pressed the = key in your last calculation. To recall this result you press the Ans key, located on the bottom row next to the = key. This means that, if you do the calculations above without any intermediate calculation, you can more simply key in:
to store this value (1591) temporarily in Ans memory.
Then key in:
Your calculator should then report the same answer as before, i.e. 1708.
Note: Further calculator guidance is provided in the solutions to some of the questions in sessions 3 and 4 of the online interactive selfassessment (for using powers, roots, logs and inverse logs keys). The study notes for Basic Statistics for PHP, STEPH and Statistics with Computing also have sections, with examples, on using the calculator.
London School of Hygiene & Tropical Medicine
BASIC MATHS SUPPORT SESSIONS
Session 1  Basic Arithmetic Exercises
You will not need a calculator for exercises 1 to 6.
1. Fractions
i) Write the following fractions in their simplest form:
12/18; 9/27; 8/16; 19/20; 12/20
ii) Write the following as improper (top heavy) fractions:
13/4; 22/8; 41/2; 31/10; 51/3
iii) Write the following as mixed fractions:
5/4; 3/2; 21/8; 21/5; 38/8
2) Work out the answers in their simplest form without using a calculator:
a) 3/4 + 4/7; b) 3/4  4/7; c) 3/4 4/7; d) 3/4 4/7; e) 4/7 3/4
3) Work out the answers in their simplest form without using a calculator:
a) 21/2 + 33/4; b) 41/3 21/2; c) 3 7/8; d) 13/4 5/8; e) 4/7 4
2. Decimals
1) Write the following decimals as fractions:
0.1 0.5 0.01 0.05 0.15 0.75 0.029
2) Write the following fractions as decimals:
14/100; 2/1000; 3/10; 4/5; 1/4
3) Write the following decimals as fractions:
1.25 3.5 7.4 2.04
4) Work out the following:
a) 2.78 100; b) 0.03 10; c) 7.3 10; d) 0.073 100; e) 0.14 100
3. Percentages
1) Change the following fractions to percentages:
3/5; 4/8; 3/10; 4/100; 1/4
2) Calculate:
20% of 180, 40% of 60, 50% of 90, 6% of 550
3) Increase:
a) 20 by 50% b) 50 by 20% c) 80 by 5%
4) Decrease:
a) 20 by 40% b) 50 by 30% c) 80 by 45%
4. Rounding
1) Round to the nearest 10:
a) 13 b) 67 c) 49
2) Round to the nearest 100:
a) 234 b) 459 c) 726
3) Round to 1 decimal place:
a) 23.45 b) 45.91 c) 3.765
4) Round the following to the nearest ten, hundred and thousand:
a) 45171 b) 20023 c) 1458.9
5) Round the following to 1, 2 and 3 decimal places:
a) 14.5171 b) 14.5175 c) 14.5871 d) 14.5875
5. Ratio
1) Write the following ratios in their simplest form:
4:6 15:18 21:7 35:25 0.3:3
2) Divide:
a) 140 in the ratio 2:3 b) 64 in the ratio 7:1 c) 130 in the ratio 2:3:5
3) i) Two numbers are in the ratio 4:7.
If the smaller number is 12, what is the larger number?
ii) Two numbers are in the ratio 5:4.
If the larger number is 20, what is the smaller number?
6. Order of operations
1) Calculate:
a) 2 + 6 12 b) (2+6) 12 c) 2 + (612)
2) Calculate:
a) (20  8) (4 + 2) b) 20  8 4 +2
3) Calculate:
((2 3) + (16 2))  ((6  3) + (21 7))
7. Applied Problems
1. There are three types of admission to a certain hospital: Ward (W), High Dependency Unit (HDU) and Intensive Care Unit (ICU). On average 2/12 of patients are admitted to HDU, and 1/18 of patients are admitted to ICU, all the rest being admitted to Ward.
a) What fraction of patients are admitted to Ward? Express your answer in its simplest form.
b) What is this fraction expressed as:
i) A decimal, rounded to 1 decimal place
ii) A percentage, rounded to the nearest percentage point.
c) On average, the nightly cost for one patient is 100 in W, 500 in HDU and 1500 in ICU. Typically, 90 patients are admitted every night divided between the units in the fractions given earlier. What will be the average total nightly cost of these admissions?
London School of Hygiene & Tropical Medicine
BASIC MATHS SUPPORT SESSIONS
Session 1  Basic Arithmetic  Further Exercises
These are intended to supplement the exercises you have already done and extend the concepts.
You will need a calculator for some of these exercises. If you need help with using your calculator please consult your calculator manual and see the information on calculator use offered in Basic Statistics for PHP or Statistics for EPH (STEPH) or Statistics with Computing.
1) a) Put the following fractions in increasing order, smallest first (hint: convert to same denominator):
17/18 3/12 1/3 7/8 5/7
b) Put the following fractions in decreasing order, largest first:
3/8 1/2 2/3 4/5 1/10
2) Work out the following:
a) (3/4 3/8) + (11/17  2/5) b) 3/4 (3/8 + 11/17)  2/5 c) 3/4 4/5 5/6
3) Change the following fractions into decimals and give the answer correct to 2 decimal places:
5/9 7/16 20/21 14/19 4/9
4) a) What percentage is 23 out of 456? Give your answer correct to 2 decimal places.
b) What percentage is 19 out of 85? Give your answer correct to 3 decimal places.
5) a) If 12% of a number is 9.6, what is the number?
b) If 40% of a number is 22.4 what is the number?
6) a) In a town 84% of the population of 50000 receive a particular vaccination. How many people
do not get the vaccination?
b) 30% of those who did not get the vaccine developed the disease. How many people got ill?
7) One of the following is not equal to 45%. Which one?
a) 45/100 b) 0.45 c) 9/20 d) 45/10
8) Calculate 75/8 13/4 42/3 .
Give your answer both as a fraction and as a decimal correct to 1 decimal place.
9) Which of the following fractions is not equivalent to 3/8?
a) 9/24 b) 6/16 c) 33/87 d) 30/80
10) Express the following ratios in their lowest form:
a) 250 g : 1.25 kg b) 35kg: 3500 g (remember 1000 g = 1 kg)
11) a) Divide 560 in the ratio 3:7:4
b) Divide 20 in the ratio 1/4 : 3/4
12) A drugs bill is divided between two drugs J and K in the ratio of 2:3 (J: K). If the smaller share is 10,000 what is the larger share?
13) 1/4 of a drugs bill is spent on drug A. 5/12 of the bill is spent on drug B and the remainder on drug C. If the total bill is 3,890,982 how much is spent on each drug?
14) A drugs bill is split between 3 drugs X, Y, Z. If 3/10 of the total is spent on drug Y and if the amount spent on drug Y is 450,528 how much is spent on the other drugs?
15) A drugs bill is split between 3 drugs X, Y, Z in the ratio 2:3:5 (X: Y: Z).
If the amount spent on drug Y is 450,528 how much is spent on the other drugs?
London School of Hygiene & Tropical Medicine
BASIC MATHS SUPPORT SESSIONS
Session 1  Basic Arithmetic  Solutions to Exercises
1. Fractions
i) The fractions in their simplest form are:
2/3; 1/3; 1/2; 19/20; 3/5
ii) The mixed fractions can be rewritten as:
7/4; 18/8 = 9/4 9/2; 31/10; 16/3
iii) The mixed fractions are:
11/4; 11/2; 25/8; 41/5; 46/8 = 43/4
2) The answers in their simplest form are:
3/4 + 4/7 = 21/28 + 16/28 = 37/28 = 19/28
3/4  4/7 = 21/28  16/28 = 5/28
c) 3/4 4/7 = 12/28 = 3/7
3/4 4/7 = 3/4 7/4 = 21/16
e) 4/7 3/4 = 4/7 4/3 = 16/21
3) The answers in their simplest form are:
a) 21/2 + 33/4 = 5/2 + 15/4 = 10/4 + 15/4 = 25/4 = 61/4
b) 41/3 21/2 = 13/3  5/2 = 26/6  15/6 = 11/6 = 15/6
3 7/8 = 21/8 = 25/8
13/4 5/8 = 7/4 8/5 = 56/20 = 14/5 = 24/5
e) 4/7 4 = 4/7 1/4 = 1/7
2. Decimals
1) 0.1 = 1/10 0.5 = 5/10 = 1/2 0.01= 1/100 0.05 = 5/100 = 1/20
0.15 = 15/100 = 3/20 0.75 = 75/100 = 3/4 0.029 = 29/1000
2) 14/100 = 0.14 2/1000 = 0.002 3/10 = 0.3 4/5 = 80/100 = 0.8 1/4 = 0.25
3) 1.25 = 11/4 3.5 = 31/2 7.4 = 72/5 2.04 = 21/25
4) a) 2.78 100 = 278 b) 0.03 10 = 0.3 c) 7.3 10 = 0.73
d) 0.073 100 = 0.00073 e) 0.14 100 = 14
3. Percentages
1) 3/5 = 3/5 100 = 300/5 = 60%
4/8 = 50% 3/10 = 30% 4/100 = 4% 1/4 = 25%
2) 20% of 180 = 20/100 180 = 36
40% of 60 = 40/100 60 = 24
50% of 90 = 50/100 90 = 45
6% of 550 = 6/100 550 = 33
3) a) 150/100 20 = 30 b) 120/100 50 = 60 c)105/100 80 = 84
4) a) 60/100 20 = 12 b) 70/100 50 = 35 c) 55/100 80 = 44
4. Rounding
1) a) 10 b) 70 c) 50
2) a) 200 b) 500 c) 700
3) a) 23.5 b) 45.9 c) 3.8
4) a) 45170, 45200, 45000 b) 20020, 20000, 20000
c) 1460, 1500, 1000
5) a) 14.5, 14.52, 14.517 b) 14.5, 14.52, 14.518
c) 14.6, 14.59, 14.587 d) 14.6, 14.59, 14.588
5. Ratio
1) 2:3 5:6 3:1 7:5 1:10
2) a) 2+3 = 5, 140 5 = 28, 28 2 = 56, 28 3 = 84
140 divided in the ratio 2:3 is 56:84
b) 7+1 = 8, 64 8 = 8, 8 7 = 56, 8 1 = 8
64 divided in the ratio 7:1 is 56:8
c) 2+3 + 5 = 10, 130 10 = 13 , 13 2 = 26, 13 3 = 39, 13 5 = 65
130 divided in the ratio 2:3:5 is 26:39:65
3) i) 4:7 = 12:?, 4 3 =12 so 7 3 = 21 and the ratio becomes 4:7 = 12: 21.
The larger number is 21.
ii) 5:4 = 20:? 5 4 = 20 so 4 4 = 16 and the ratio becomes 5:4 = 20:16.
The smaller number is 16.
6. Order of Operations
1) a) 2 + 6 12 = 2 + 72 = 74 b) (2+6) 12 = 8 12 = 96
c) 2 + (612) = 2 + 72 =74
2) a) (20  8) (4 + 2) = 12 6 = 2 b) 20  8 4 + 2 = 20  2 +2 = 20
3) ((2 3) + (16 2))  ((6  3) + (21 7)) = (6 + 8) (3 + 3) = 14 6 = 8
7. Applied Problems
1) a) Fraction admitted to HDU and ICU is 2/12 + 1/18 = 1/6 + 1/18 = 3/18 + 1/18 = 4/18. So
fraction admitted to W is 1 4/18 = 14/18 = 7/9.
b) 7/9 = 0.7777... = i) 0.8 to 1 decimal place ii) 78% to nearest % point.
c) 1/6 x 90 = 15 HDU patients; 1/18 x 90 = 5 ICU patients; remainder = 70 W patients.
So typical nightly cost is
(15 x 500) + (5 x 1500) + (70 x 100) = 7500 + 7500 + 7000 = 22000.
London School of Hygiene & Tropical Medicine
BASIC MATHS SUPPORT SESSIONS
Session 1  Basic Arithmetic Further Exercises  Solutions
1) a) Increasing order, smallest first: 3/12 1/3 5/7 7/8 17/18
b) Decreasing order, largest first: 4/5 2/3 1/2 3/8 1/10
2) a) (3/4 8/3) + (5534/85) = 24/12 + 21/85 = 2 + 21/85 = 221/85
b) 3/4 (3/8 + 11/17)  2/5 = 3/4 (51/136 + 88/136)  2/5 = 3/4 139/136  2/5
= 3/4 136/139  2/5 = 102/139  2/5 = 510/695  278/695 = 232/695
1/2
3) 0.56 0.44 0.95 0.74 0.44
4) a) 23/456 100 = 5.04 (converting the fraction 23/456 to a decimal)
b) 19/85 100 = 22.353 (converting the fraction 19/85 to a decimal)
5) a) 12% is 9.6 so 1% is 9.6/12 and 100% is 9.6/12 100 = 80
b) 40% is 22.4 so 1% is 22.4/40 and 100% is 22.4/40 100 = 56
6) a) 16% do not get the vaccine. 16/100 50000 = 8000
b) 30% of 8000 get ill. 30/100 8000 = 2400
7) d) is not equal to 45%
8) 75/8 13/4 42/3 = 61/8 7/4 14/3 = 61714/843 = 5978/96 = 2989/48 = 62.3
9) c) 33/87 does not equal 3/8
10) a) 250 g : 1.25 kg = 250 g : 1250g = 25:125 = 1:5
b) 35kg: 3500 g = 35000 g :3500 g = 350:35 = 10:1
11) a) ratio 3:7:4 so take 3+7+4 = 14
find 3/14, 7/14 and 4/14 of 560 which is 3/14 560, 7/14 560 and 4/14 560
so 560 divided in the ratio 3:7:4 is 120:280:160
b) ratio 1/4 : 3/4 is same as ratio 1:3
1/4 20 = 5 3/4 20 = 15 so 20 divided in the ratio 1/4 : 3/4 is 5:15
12) 2/3 = 10000/? So the larger share is given by 3/2 10000 = 15000
13) 1/4 3,890,982 = 972745.50 5/12 3,890,982 = 1621242.50
3,890,982 (972745.50 +1621242.50) = 1296994
14) 3/10 of the total is 450,528 (the amount spent on drug Y)
so 1/10 of the total is 450,528/3
so 10/10 (the whole) of the total is 450,528/3 10 = 1501760
so the amount spent on the other drugs is 1501760  450,528 = 1051232
15) Splitting the bill according to the ratio 2:3:5 is equivalent to taking 2/10 , 3/10 , 5/10 so
3/10 of the total is 450,528 (the amount spent on drug Y) and this is the same as question 14.
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