Self assessment ,f
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Application of Number
Level 1
Contents
Exercise 1 3
Millions 4
Exercise 2 5
Addition 8
Subtraction 9
Multiplication 10
Division 11
Adding Decimals 12
Subtracting Decimals 15
Multiplying Decimals by Whole Numbers 19
Dividing Decimals by Whole Numbers 21
Multiplying, dividing by powers of 10 24
Rounding Off and Approximation 25
Rounding numbers to the nearest 10 26
Rounding numbers to the nearest 100 27
Rounding numbers to the nearest 1000 28
Rounding 29
Rounding off amount of money 30
Handling Data 31
Charts 31
Graphs 31
Line graphs 32
Pie charts 32
Frequency tables 33
Conversion graphs 33
Mean and Range 34
Range 37
Averages (1) 40
Working out equivalent ratios 41
Writing ratios in common units 42
Ratio 44
Proportion 45
Comparing Lengths 46
Converting Between Units 50
Conversions between Metric and Imperial Systems 51
Conversions between Metric and Imperial Systems 52
Distances 53
Areas and Volumes 55
Area of rectangles 56
Area 58
Volume 60
Comparing fractions and percentages 61
Fractions, Decimals and Percentages 62
Finding fraction parts 64
Percentages 68
Exercise 1
1) Write these numbers in words:
a) 6 320
b) 12 503
c) 163 900
d) 268 726
e) 402 150
2) Write these numbers in figures:
a) three thousand, four hundred and twenty
b) ninety six thousand, one hundred and thirty one
c) four hundred and twenty five thousand, seven hundred and two.
d) eight hundred and two thousand, and ninety three.
e) seventy thousand and sixteen.
f) two hundred and five thousand, and six.
For the following questions, select the appropriate answer A, B, C or D
3) The largest ever football crowd was 199 000 people. How is this number written in words ?
a) One hundred and ninety-nine million
b) One million nine hundred thousand
c) One hundred and ninety-nine thousand
d) Nineteen thousand nine hundred
4) Twenty thousand five hundred and six people went to a concert.
The number of people was
a) 2 056
b) 2 506
c) 20 506
d) 205 006
Millions
If you're talking millions, you need to have at least seven figures in there.
Examples
1) One million written in figures is
1 000 000
2) Two million, three hundred and forty two thousand
Write in the two, leave a space then write in three hundred and forty two as you normally would.
Leave another space then write in three zeros to show that there are no hundreds, tens or units.
2 342 000
3) Four million, two hundred and five thousand, one hundred and eighty
Write in four, leave a space then write in two hundred and five. Leave another space then write in one hundred and eighty.
4 205 180
2 500 000
Write in two then leave a space.
Whatever comes after the decimal point, in this case five, write it in and follow it with five zeros so that there are seven figures in total.
Exercise 2
This is really boring but excellent practice I
1) Write these numbers in figures:
a) five million
b) nine million
c) two million, five hundred thousand
d) four million, two hundred thousand
2) Write these numbers in words:
a) 2 000 000
b) 4 000 000
c) 5 500 000
3) Write these numbers in figures:
a) one million, two hundred and twenty two thousand
b) nine million, four hundred and four thousand
c) five million, forty two thousand
4) Write these numbers in words:
a) 5 757 000
b) 2 120 000
c) 1 304 000
d) 6 045 000
e) 7 002 000
5) Write these numbers in figures:
a) six million, one hundred and seventy five thousand, two hundred and forty six
b) five million, three hundred and four thousand, two hundred and six
c) one million, eight thousand, one hundred and twenty
6) Write these numbers in words:
a) 1 234 567
b) 4 460 020
c) 2 009 009
d) 3 040 040
7) How much are the following amounts in figures ?
a) £8.5 million
b) £2.9 million
c) £0.5 million
For the following questions, circle either answer A, B, C or D
1) A total of four hundred and three thousand, five hundred and two people in Greater Manchester voted at the last general election. How many people is this in figures ?
a) 43 502
b) 430 502
c) 403 502
d) 4 003 502
2) A company made £308 010 profit in one month. How much is this amount in words ?
a) three million, eighty thousand and ten pounds
b) three hundred and eight thousand and ten pounds
c) thirty eight thousand and ten pounds
d) thirty thousand, eight hundred and ten pounds
3) The population of Cairo was nine million, five hundred thousand, four hundred and twenty when last counted. How do you write this number in figures ?
a) 9 500 000 420
b) 950 420
c) 95 420
d) 9 500 420
4) Jenny was lucky enough to win £6.2 million on the National Lottery. How much is this in figures ?
a) £620 000
b) £6 200 000
c) £62 000
d) £62 000 000
You can "use" numbers in four different ways..
|WAYS |SIGNS AND WORDS USED |
|Addition |+ adding, sum of, total |
|Subtraction |- subtract, take-away, difference between, minus |
|Multiplication |x multiply, "times", product |
|Division |( divide, sharing |
SEE IF YOU CAN DO THESE!
Addition
1) 76
+ 18
2) 89
+3
3) 625
+ 227
4) 206
498
+ 6
5) 625
+398
6) 462 + 87
7) Add these numbers; 274; 648
8) Find the total of 362 and 479
9) Find the sum of 462, 87 and 8
10) Add two hundred and seven, to six hundred and ninety four.
Subtraction
1) 462
- 120
2) 865
- 626
3) 300
- 212
4) Subtract 623 from 900
5) What is the difference between 174 and 310?
6) From 436 take 208
7) How many more than 297 is 642?
8) How many less than 700 is 285?
9) Seven hundred and four minus eighty nine.
Multiplication
1) 8 x 7
2) 24
x 2
3) 164
x 4
4) 208 x 8
5) 412
x 26
6) Multiply 245 by 4
7) Double four hundred and eighty six
8) Which number is 3 times greater than 297?
9) What is the product of 6 and 4?
10) 325 times 16
Division
1) 8 ( 4
|2) |4 |48 |
|3) |5 |365 |
|4) |5 |505 |
|5) |26 |1924 |
6) Divide 707 by 7
7) How many 6's are there in 912?
8) Divide nine hundred and forty-five by 9
9) Divide five thousand, five hundred and fifty-five by five.
10) 164
4
Adding Decimals
When you add money you are adding decimals
|( |
|£1.75 + |
|£2.64 |
|£4.39 |
| 1 ( |
You can see that the decimal point is separating the pounds from the pence.
All the decimal points are in a line.
This is the main thing to remember when adding decimals
Keep the decimal points in a column.
1) £2.35+
£1.40
2) £27.99+
3.67
3) £102.70+
23.56
4) £1.35 + £3.45
5) £28.53 + £3.55 + £7.00
6) Katie goes to Kwik Save and buys Sausages for £1.39, Cheese for £2.50 and Apples for £1.27. How much did she spend?
Have a look at this example.
|( |
|3.76 |
|7.51 |
|11.27 |
| 1 ( |
You add and carry numbers in just the same way as when using whole numbers or money.
7) 3.24+
6.51
8) 5.25+
6.19
9) 11.35+
1.17
10) 5.75 + 16.89 + 4.81
11) Add up 14.67 metres and 3.09 metres
12) Karen's yard is 3.45 metres long and 2.57 metres wide. How far is it all around the edge (the perimeter)?
Sometimes it helps if you fill the spaces at the end of each number with zeros (0's)
| | | ( |
|3.76 + | | 3.760 + |
|7.5 | | 7.500 ( |
|0.375 | | 0.375 |
|11.635 | | 11.635 |
It doesn't make any difference to the answer. It just makes it easier to add up without losing your place.
Only put zeros at the end of a number (not in the middle)
13) 3.25 +
1.5
14) 25.3 +
16.97
15) 10.5 + 0.375
16) 19.9 + 10.101 + 0.009
17) 0.425 + 7.6 + 3.48
18) Paula is making curtains. Her window is 1.7 metres high, and she wants the curtain 0.15 metres longer than the window. How long will the curtain be?
Sometimes you will have to add whole numbers to decimal numbers. To do this put a decimal point at the end of the whole number and then line it up with the rest of the sum.
For example: 23 + 2.35
23. + Put a decimal point at the end of the
2.35 23 and line it up.
23 00 + You can Put zeros In to make it look more
2 35 balanced. We see this in money ….
25 35 £23 is the same as £23.00
19) 25 + 3.33
20) 102.5 + 25 + 3.4
21) 0.785 + 785
22) 2.92 + 0.292 + 292
23) There are 7 litres of water in the urn. Jacob adds 15.5 litres more. How much is in there altogether?
24) Yasmin is fitting a skirting board in her bedroom. The walls measure 3 metres, 1.6 metres, 1.75 metres, 3 metres, 0.95 metres and 0.85 metres. How much skirting board will she need?
Remember Line up your decimal points
Carry numbers as usual
You can use zeros at the end of each number if you want
Line up whole numbers with the point at the end
Subtracting Decimals
When you take away money you are subtracting decimals.
|( |
|£3.75 + |
|£2.54 |
|£1.21 |
| ( |
You can see that the decimal point is separating the pounds from the pence.
All the decimal points are in a line. Always make sure the biggest number is on top
This is the main thing to remember when taking away decimals
Keep the decimal points in a column.
1) £2.75 -
£1.40
2) £27.99 -
£3.67
3) £162.78 -
£23.56
4) £16.35 - £3.45
5) £28.53 - £3.55
6) Arthur gets £67.56 a week. He spends £12.45 at the supermarket. What has he got left?
7) Bashir spends £3.35 at the newsagent. He pays with a £5.00 note. How much change will he get?
8) Jenny buys a bed for £199.99. She gets £19.98 discount. How much will she pay now?
Have a look at this example.
13.76 -
7.51
6.25
You subtract and borrow numbers in the same way as when using whole numbers or money.
9) 9.24 -
6.51
10) 17.25 -
6.19
11) 11.35 -
1.17
12) 16.89 - 4.91
13) 14.67 - 9.09
14) Yasmin has 3.7 metres of material. She needs 1.8 metres to make a dress. How much will she have left? Will she have enough material left to make another dress for her sister?
Sometimes it helps if you fill the spaces at the end of each number with zeros (O's)
9.76 - 9.76 -
7.9 7.90(
1.86 1.86
It doesn't make any difference to the answer. It just makes it easier to take away without losing your place and helps when you are borrowing.
6.1 - 6.10 -
3.45 3.45
2.65 2.65
Only put zeros at the end of a number (not in the middle)
15) 3.25 - 1.5
16) 25.3 - 16.97
17) 10.52 - 0.375
18) 19.92 - 10.101
19) 7.666 - 3.4
20) Janet is going to drive to Manchester. It is 34.9 miles. She stops after 13.0 miles to get some petrol. How much further does she have to go?
21) On Monday you can get 1.7 dollars for a pound. By Friday you get 1.61 dollars. What is the difference?
22) James and Edna share a bottle of wine (0.75 of a litre). Edna drinks 0.4 litres. How much did James drink?
Sometimes you will have to take away decimal numbers from whole numbers. To do this put a decimal point at the end of the whole number and then line it up with the rest of the sum. You can then add zeros (like you did above)
For example:23 - 2.35
23. -
2.35
Put a decimal point at the end of the 23 and line it up.
23.00
2.35
20.65
You can put zeros in to make it look more balanced. Take away as usual.
23) 102 - 25.4
24) 5.85 - 2
25) 2 - 0.292
26) 7.32 - 3
27) 27 - 3.3
28) Fiona buys a carpet which measures 7 metres. Her hall is 5.25 metres long. How much carpet has she got left? Will she have enough to carpet the landing which is 1.85 metres long?
29) Jamal can swim 2 lengths in 50.2 seconds. How much under a minute (60 seconds) is this?
30) A jug holds 2 litres. Lee pours a glass of juice which holds 0.125 litres.
How much is left in the jug?
How much is left if he pours himself another glass?
Remember
Line up your decimal points
Borrow numbers as usual
You can use zeros at the end of each number if you want.
Line up whole numbers with the point at the end.
Multiplying Decimals by Whole Numbers
When you have your answer check it by estimating and then with a calculator
1) £5.75 x 6
2) £9.40 x 7
3) £9.99 x 2
4) £107.44 x 5
5) Jan bought 6 towels. They were £4.99 each. How much did she spend altogether?
6) Dave went to buy Christmas cards. He got 7 packs at £3.15 each. How much did he spend?
7) 1.1 x
9
8) 5.4 x
2
9) 0.51 x
3
10) 107.42 x
5
11) 17.5 x
6
12) 0.999 x
2
13) Chris is building a fitted kitchen. He needs 5 units measuring 52.5cm and 3 units measuring 76.25 cm. How much does this come to altogether?
14) Cath is baking Christmas cakes for her family. She needs to make 4 cakes. For each one she needs 1.45 kg of dried fruit, 0.5 kg flour and 0.225 kg of almonds. How much of each ingredient will she need altogether?
Dividing Decimals by Whole Numbers
When we divide money we are dividing decimals.
If we share £5.16 by 3
| | 1 |
|3 |£5.16 |
| | 1 |
|3 |£5.216 |
You carry the remainder 2 onto the next number as usual.
| | 1.7 |
|3 |£5.216 |
When you meet the decimal point in the sum, put it up in your answer. Now carry on dividing 3 into 21
| | 1.72 |
|3 |£5.216 |
Finish off the sum and you have the answer. Does it look about right?
Have a go at dividing these sums of money.
1)
| | |
|6 |£8.40 |
2)
| | |
|7 |£2.24 |
3)
| | |
|2 |£39.84 |
4)
| | |
|8 |£16.56 |
5) Sharon buys 5 mugs for £9.95. How much is each one worth?
6) Eddie's electricity bill comes to £142.40 a year. How much is this each quarter (every three months)?
7) It costs £62.50 for 5 driving lessons. How much does 1 cost?
Now try these yourself.
8)
| | |
|5 |9.175 |
9)
| | |
|57 |2.8 |
10)
| | |
|2 |7.156 |
11)
| | |
|3 |10.2 |
12)
| | |
|6 |6.390 |
13)
| | |
|9 |70.38 |
Yasmin has a piece of wood measuring 1.72 metres. She wants to put 2 shelves in her hall. How long will each shelf be?
Evelyn has 7.36 metres of material. She wants to make 4 curtains out of it. Will she have enough material to make them if each one needs to be 1.8 metres long?
Sometimes you might get to the end of the sum and still have a number to carry (a remainder).
For example:
| |1 |
|4 |6.22 |
| |( |
Start as normal
| |1.5 |
|4 |6.22 |
| | ( |
We divide 22 by 4, but still have a remainder of 2
| |1.5 5 |
|4 |6.2220 |
| | ( |
Add a 0 on the end and carry on as usual, until finished
You can carry on adding 0s for as long as you want, but usually it makes sense to ROUND OFF after 2 or 3 decimal places.
Multiplying, dividing by powers of 10
0.25 x 10 = 2.5, 3.197 x 100 = 319.7. In order to multiply by a power of ten, the digits must
be moved one place to the right (making the number bigger) for every zero in the divisor
(that is one place to multiply by 10, two places for 100 etc.)
6 ( 10 = 0.6, 8 ( 100 = 0.08. In order to divide by a power of ten, the digits must be moved
one place to the left for every zero in the divisor (that is, two places to divide by 100, three places for 1000 etc.)
1) 8.25 x 100 =
2) 4.927 x 1000 =
3) 0.05 x 10 =
4) 18.398 x 100 =
5) 6.2 x 1000 =
6) 9.41 x 100 =
7) 84 (10 =
8) 60 ( 100 =
9) 8.4 ( 1000 =
10) 27 ( 100 =
11) 1.06 (10 =
12) 2.41 ( 100 =
Rounding Off and Approximation
Example 1
The number 23.62 rounded to the nearest whole number
23.62 is post half way so it is closest to 24.
Example 2
The number 23.62 rounded to one decimal place
23.62 is before half way so it is closest to 23.6.
Draw pictures to help you round the following numbers to the accuracy
given in each question.
1) 43.6 to the nearest whole number
2) 3.57 to one decimal place
3) 8.61 to one decimal place
4) 132 to the nearest 100
5) 58.99 to the nearest 10
6) 63.49 to one decimal place
7) 63.49 to the nearest whole number
8) 8.55 to one decimal place
Rounding numbers to the nearest 10
Round the following to the nearest ten:
Example: 46 is between 40 and 50 and would be rounded to 50.
|43 is between | |and | |and would be rounded to | |
|57 is between | |and | |and would be rounded to | |
|72 is between | |and | |and would be rounded to | |
|85 is between | |and | |and would be rounded to | |
|90 is between | |and | |and would be rounded to | |
|103 is between | |and | |and would be rounded to | |
1) A company made a profit of £7 821 in one year. What is this figure to the nearest £10?
2) A theme park had 9,462 visitors over a summer period. What is this to the nearest ten?
Rounding numbers to the nearest 100
Round these numbers to the nearest hundred:
Example: 231 is between 200 and 300 and would be rounded to 200,
|384 is between | |and | |and would be rounded to | |
|946 is between | |and | |and would be rounded to | |
|162 is between | |and | |and would be rounded to | |
|853 is between | |and | |and would be rounded to | |
|211 is between | |and | |and would be rounded to | |
|7,423 is between | |and | |and would be rounded to | |
1) The cost of decorating a school was estimated at £25 670. What is this to the nearest £100?
2) Temi won £12 478 on the National Lottery. What is this figure to the nearest hundred pounds?
Rounding numbers to the nearest 1000
Round these numbers to the nearest thousand:
Example: 7 852 is between 7 000 and 8 000 and would be rounded to 8 000.
|1,963 is between | |and | |and would be rounded to | |
|2,243 is between | |and | |and would be rounded to | |
|4,625 is between | |and | |and would be rounded to | |
|8,463 is between | |and | |and would be rounded to | |
|9,271 is between | |and | |and would be rounded to | |
|19,253 is between | |and | |and would be rounded to | |
1) A distance between two countries is 12 870 km. What is this distance to the nearest 1000 km?
2) A newspaper sold 56 792 copies. What is this to the nearest thousand copies?
Rounding
1) Round the following football attendances to the nearest thousand:
17, 394
19, 801
35, 009
2) Round the following mileages to the nearest 10 miles:
369 miles
805 miles
466 miles
3) Round the following sums of money to the nearest £100:
£650
£8720
£299
4) Round the following weights to the nearest 10kg:-
637kg
3059kg
309kg
5) Round the following to the nearest £:-
£36.75
£6.99
£55.55
Rounding off amount of money
To the nearest pound
Remember that if the number after the decimal point is five or more, round up to the next pound.
Examples
£2.56 rounds up to £3.00
£8.49 rounds down to £8.00
To the nearest ten pence
Examples
£2.56 rounds up to £2.60 as 56p is closer to 60p.
£8.44 rounds down to £8.40 as 44p is closer to 40p.
Self assessment
Round these amounts of money off to the nearest pound.
£1.35
£2.89
£19.99
£111.49
£205.51
£3.50
Round the following amounts to the nearest £0.10 or 10p
£ 6.52
£ 13.18
£271.65
£215.55
£205.51
£989.89
Handling Data
• Data is information. Interpreting data means working out what information is telling you.
• Information in newspapers, on television, in books and on the internet is sometimes shown in charts, tables and graphs.
• It is often easier to understand the information like this rather than in writing, but it is important to read all the different parts of the graph or chart.
Charts
The title tells us what the chart is about
The column headings tell us what data is in each column:
• the name of the bike
• what colour it is
• its price
• and how many gears it has
You can use the chart to find out information about each bike by looking at each row in turn
• The Ranger is silver and has 5 gears
Bikes sold this week
|Name |Colour |Price |Gears |
|Ranger |Silver |£140 |5 |
|Outdoor |Red |£195 |10 |
|Tourer |Blue |£189 |15 |
|Starburst |Silver |£215 |15 |
|Mountain |White |£129 |5 |
Graphs
Graphs come in many different styles
Bar graph
[pic]
Tip: With any graph, always look carefully at
• the title
• the scale
• the axis headings
Line graphs
Are made by joining the tops of bar-line graphs. This can make it easier to look at the shape of the graph.
This line graph shows that the temperature is falling each hour.
Pie charts
Pie charts are circular, like a pie! Each section of the pie shows a fraction of a total amount. This pie chart shows where 40 went on their last holiday.
One quarter of the people went to Europe. That means 10 people (40 ( 4) people went to Europe.
The UK was the most popular holiday destination.
Can you work out the second most popular?
Frequency tables
A frequency table shows information about a set of data. Sometimes there is so much data that the only way to show it all is to put it into groups called intervals.
This graph shows the heights of a class of children. The heights are grouped in equal intervals of 5cm. This means that 1.30 -1.34 includes children with heights of 1.30m, 1.31m, 1.32m, 1.33m
[pic]
How many children are between the heights of 1.45m and 1.49m?
Conversion graphs
Conversion graphs are used to change one set of values to another.
This graph converts centimetres to inches. 5cm is approximately 2 inches.
[pic]
Approximately how many centimetres are equal to 5 inches?
Mean and Range
After completing this unit, you will be able to:
• calculate the mean average for up to ten items of data
• calculate the range of up to ten items of data
Mean Average
The average value of a group of numbers is a number that can be used to represent the
whole group.
The MEAN AVERAGE is calculated as follows:
• Add up all the numbers.
• Divide your answer by the
• Number of numbers there are
Examples
1) Here are five numbers.
2 11 8 6 3
Find the mean number.
Answer
2
11
8
6
3 +
30
there are five numbers, so divide 30 by 5.
30 ( 5 = 6 so the mean is 6.
2) The temperature of a science laboratory was recorded daily at noon over six days. The results are given in the table below.
|Monday |Tuesday |Wednesday |Thursday |Friday |Saturday |
|9°C |11°C |14°C |12°C |8°C |0°C |
What is the mean temperature at noon ?
Answer
9
11
14
12
8
0 +
54
2
there are six days so divide 54 by 6
54 ( 6 = 9 so the mean temperature is 9°C.
3) Find the mean of the following groups of numbers.
a) 12 4 14 3 7
b) 4cm 3cm 2cm 2cm 8cm 4cm 4cm 3cm
c) £2.50 £1.50 £1.00 £1.60
4) Sharon made a record of how much she spent each week on entertaining her daughter. What is the mean amount she spends per week ?
|Week 1 |Week 2 |Week 3 |Week 4 |Week 5 |Week 6 |
|£18 |£10 |£15 |£9 |£12 |£8 |
a) £10
b) £12
c) £15
d) £13
5) The table below shows the daily hours of sunshine in the Costa del Sol for the months of January to September.
|Jan |Feb |Mar |Apr |May |Jun |Jul |Aug |Sep |
|5 |7 |8 |9 |10 |11 |12 |10 |9 |
What is the mean of the daily hours of sunshine for the months shown?
a) 5
b) 9
c) 12
d) 81
6) As part of a Health and Social Care assignment, a student researched the price of ten different drinks for children, five fizzy drinks and five fruit juices. Her task was to find out if, on average, it is cheaper to buy fizzy drinks or fruit juices.
|Fruit juice |Price per 300 ml |Fizzy drink |Price per 300 ml |
|A |50p |A |55p |
|B |65p |B |85p |
|C |80p |C |55p |
|D |60p |& |65p |
a) What's the mean price of fruit juice ?
b) What's the mean price of fizzy drink ?
c) What should the student's conclusion be ?
Range
The range of a group of numbers tells you how wide the numbers
are spread.
The RANGE is calculated as follows:
THE BIGGEST VALUE MINUS THE SMALLEST VALUE.
Examples
Find the range of the following numbers.
5 7 2 9 8 6 12 4
Answer
The biggest number is 12.
The smallest number is 2.
Range = 12 - 2 = 10.
The temperature of a science laboratory was recorded daily at noon over six days. The results are given in the table below.
|Monday |Tuesday |Wednesday |Thursday |Friday |Saturday |
|9°C |11°C |14°C |12°C |8°C |0°C |
What is the range of noon temperatures for the six days ?
Answer
The smallest number is now 0°C.
14 - 0 = 14, so the range is 14°C.
1) Find the range of the following groups of numbers.
a) 9 15 1 8 2 3
b) 10oc 2oc 22oc 15oc 7oc 2oc
c) £3.00 £1.20 £4.50 £6.30 £2.00 £9.10
2) Mike is a car salesman. The table shows how many cars he has sold each month for a six month period
|April |May |June |July |August |September |
| |10 |5 |12 |7 |8 |
What is the range of number of vehicles he has sold from April to September
a) 5
b) 7
c) 9
d) 12
For the following questions, circle either answer A, B, C or D.
Five friends take part in a sponsored run and record the amounts they collected. The amounts are recorded in the table
|Name |Ada |Bal |Cath |Dan |Ernest |
|Sponsorship (£) |10.00 |24.00 |23.50 |42.50 |60.00 |
Use the information above to answer questions 1 and 2.
1) What was the mean amount of sponsorship money collected per person ?
A £160 B £35 C £30 D £32
2) What was the range of the sponsorship money to be collected?
A £60 B £50 C £10 D £35
Willy is worried about his spending and decided to make a note of how much he withdraws from the cash machine on a daily basis. Over the past ten days he has made the following withdrawals:
£50 £40 £40 £140 £70
£0 £50 £90 £130 £50
Use this information to answer questions 3 and 4.
What is the mean amount he has withdrawn ?
A £66 B £70 C £50 D £140
What is the range of Willy's withdrawals ?
A £100 B £66 C £140
Averages (1)
1) Find the average (mean) of 21, 17, 8, 3, 22
2) What is the total of the following mechanics' monthly wages:- £585, £836, £624, £453?
3) What is the mechanics' average wage?
4) What is the range of their wages?
5) The following distances are travelled on 40 litres of fuel by a fleet of taxis:-
56km, 87km, 98km, 72km, 86km, 76km.
What is the average length of journey?
6) What was the average fuel consumption in question 5?
7) The time taken by 6 mechanics to do a particular job was:-
10min, 13min, 15min, 17min, 10min, 9min.
What was the average time taken?
8) If I travel 300km in 3 hours, what is my average speed?
9) During a period of 9 months a car travels 7938km. What is the average distance travelled per month?
10) Five mechanics earn the following amounts as a weekly wage:- £98, £104, £106, £101, £96.
What is their average weekly wage?
What is the range of their wages?
Working out equivalent ratios
To make double this amount of pink paint you would need 8 white tins to 2 red tins.
This would give a ratio of 8:2
These two ratios are called equivalent ratios.
You can multiply or divide both parts of a ratio by any number to give an equivalent ratio. Both 8 and 2 can be divided by 2, so 8:2 is the same as 4:1
Simplifying ratios
e.g. Natasha and Rachel share a flat but because Natasha has a larger room they agree
to share the rent in the ratio 60 : 40. This can be simplified as follows:-
Divide both sides by 10 to get 6 : 4
Divide both sides by 2 to get 3:2
There are no further common factors of the two sides so this is the simplest form of the
ratio.
Self Assessment One - Writing ratios in their simplest form
Write each ratio in its simplest form
1) 6:2
2) 18 : 24
3) 80 : 20
4) 12 : 48
5) 12 : 25
6) 14 : 63
7) 75 : 25
8) 42 : 63
9) 4 : 8 : 6
10) 30 : 90 : 120
11) 14 : 28 : 35
12) 21 : 63 : 105
13) The label for a bottle of floor cleaning liquid says that one unit of fluid needs to be mixed with 20 units of water.
a) How much water is mixed with 5 units of cleaning fluid?
b) How much cleaning liquid is mixed with 180 units of water?
Converting to similar units
It is important to remember that when amounts are given in different units they must first
be converted into a common unit before they can be expressed as a ratio.
They can then be put into the simplest form.
E.G. Two TV programmes last 45 minutes and 2 hours.
Express the times of the two programmes as a ratio.
The times are 45 minutes : 2 hours
Convert both times into minutes 45 minutes : 120 minutes
This is now a ratio of 45 : 120
Divide both by 15 3 : 8
This is the ratio in its simplest form.
Writing ratios in common units
Write each ratio in a common unit and then put it into its simplest form.
1) 20 minutes : 3 hours
2) 20mm : 3cm
3) 3kg : 850g
4) £3 : 60p
5) 250 ml : 1.75 litres
6) 2500m : 5km
7) An Orange drink is mixed in the ratio of 25 ml of orange cordial to 1 litre of water.
a) How much of this cordial would you mix with 4 litres of water?
b) What is the correct mixing ratio of cordial to water?
Write this ratio in its simplest form.
Ratio
Simplify the following:-
1) 3:6
2) 42 : 49
3) 40:50
4) 175 : 200
5) 50:60:70
6) 5000 : 7500
Simplify the following:
7) 500cm : 1m
8) £1.50 : 50p
9) 30p : 90p
10) 5 .5cm : 20mm
11) 2 kg : 500g
12) £12.50 : £2.50
13) If 2kg potatoes cost 32p, what would 7 kg cost?
14) 6 plates cost £7.20. What do 11 cost?
15) A machine makes 16 articles in half an hour. How many will it make in
a) 1 hour
b) 4 hours
c) 6 hours?
Proportion
1) 3kg potatoes cost 45p. What do 10kg cost?
2) 8 bolts cost 40p. What do 24 cost?
3) A model car is 50 times smaller than a real car.
a) What length on the real car is represented by 4 cm on the model?
b) 250cm on the real car is represented by how much on the model?
4) A mechanic earns £15 in 3 hours. Find how much she earns in
a) 1/2 hour
b) 4 hours
c) 7 hours
5) A mixture of paint used for spraying a car was made up of 2 litres blue with 6 litres grey. What is this, as a ratio, in the lowest terms?
6) Divide £50 in the ratio 1: 4
7) 4 lightbulbs cost £3.00. What would 5 cost?
8) 3 cans of cola cost £1.20. What would 5 cost?
Comparing Lengths
Sometimes you will have to compare different lengths and different units.
They become easier to compare once you have converted them to the same units.
You can convert any metric measurements like this:
Kilometres (km)
[pic]
Millimetres (mm)
Using this chart change:
1) 9cm to mm
2) 21.5cm to mm
3) 30mm to cm
4) 95mm to cm
5) 4m to cm
6) 2.5m to cm
7) 400cm to m
8) 725cm to m
9) 7km to m
10) 10.675km to m
11) 5000m to km
12) 13cm to mm
13) 10.9cm to mm
14) 151mm to cm
15) 9mm to cm
16) 17m to cm
17) 19.3m to cm
18) 950cm to m
19) 87cm to m
20) 7.5km to m
21) 0.725km to m
22) 9990m to km
Which is bigger - 205cm or 2.5m?
We know 2.5m is the same as 250cm, so 2.5m is the longer length
Which is bigger?
1) 105cm or 1m
2) 10mm or 1.1cm
3) 700cm or 7m
4) 105mm or 1m
5) 100mm or 15cm
6) 3555m or 3.9km
7) 1.275m or 0.3km
8) Match up the measurements in column A with the one that is the same length in column B
|A |B |
|30cm |7.6cm |
|9.5cm |300mm |
|107cm |950cm |
|3mm |0.237km |
|9.5m |3000mm |
|237m |760cm |
|76mm |0.3cm |
|7.6m |1.07m |
|3m |95mm |
|237cm |2.37m |
Now try and use these skills to work out these perimeters.
1) Leanne measures her front room.
It is 6 metres long and 350 centimetres wide.
What is its perimeter?
2) She decides to extend the room.
When it is finished it is still 6m long but it is 150cm wider.
What is the new width and the new perimeter?
3) Leanne also has a bedroom with these measurements:
a) What are the missing measurements?
b) What is the perimeter of the room?
c) Will she be able to fit her furniture along the wall measuring 2.6m?:
bed (width 1.5m)
bedside table (45cm)
chest of drawers (70cm)
Converting Between Units
Length
1cm = 10mm
1m = 10cm
1km = 1000m
Weight
1kg = 1000g
Capacity
1litre=1000ml
How many millimetres are there in:
8cm
22cm
36cm
85cm
50m
How many centimetres are there in:
5m
28m
36m
90m
150m
How many metres are there in:
2km
12km
25km
65km
100km
How many grams are there in:
4kg
10kg
40kg
60kg
100kg
How many kilograms are there in:
6000g
15000g
25000g
82000g
10000g
Conversions between Metric and Imperial Systems
1) Convert the following to centimetres.-
a) 3 metres
b) 3.6m
c) 36mm
d) 0.065km
e) 5inches
f) 3.4ft
2) Convert the following to metres:-
a) 40cm
b) 213mm
c) 1.39km
d) 27inches
e) 5yards
3) Convert the following to grams:-
a) 300mg
b) 15.34kg
c) 12.31b
d) 0.231b
e) 17oz
4) Convert the following to litres. -
a) 3 gallons
b) 5 pints
5) (1 mile - 1.6km, 1km - 0.6miles)
Convert the following:-
a) 14 miles to kilometres
b) 26km to miles
6) 1 lb = 16oz
Convert the following to ounces:-
a) 2 1/2 lb
b) 1/2 lb
c) 2 lb 3oz
Conversions between Metric and Imperial Systems
|1 inch - 25mm |1 ounce - 28g |
|1 foot - 30cm |1 pound (lb) - 450g |
|1 yard - 90cm |1 cwt (hundredweight) - 50kg |
|1m - 39 inches |100g - 3.5oz |
|(3ft 3in) or 3.25ft |250g - 8.8 oz |
|25cm - 10 inches | |
|1 mile - 1.6 km |1kg - 2.2 lb |
| |1 tonne - 1 ton |
Complete the following:-
1) 600g = oz
2) 75cm = inches
3) 3 lb = g
4) 5 miles = km
5) A vehicle is travelling at 70 miles per hour. How many km/hr is this?
6) A mechanic weighs 80kg. How many pounds is this?
7) Write as a fraction of a pint in lowest terms :-
a) 6fl.oz
b) 15fl.oz
c) 8 fl.oz
8) Convert the following, if 1 fluid ounce = 30mh-
a) 2fl.oz= ml
b) 5 floz. = ml
c) 1/3 floz. = ml
Distances
Try these questions and see how you do
Check each of these answers and tick one of the boxes.
| |True | |False |
|A map has a scale of 2 cm = 1km, so 4cm = 1/2 km | | | |
| | | | |
|On a map with a scale of 3cm = 1 mile, 12 cm represents 4 miles | | | |
| | | | |
|The distance between two towns is measured as 12cm. the map uses a scale of 4cm to 1km so the towns | | | |
|are 3km apart | | | |
| | | | |
|A map uses a scale of 5cm to 1mile and the distance between two towns on the map is 20cm. this | | | |
|represents 100miles | | | |
Use the map below to answer the questions. (Scale 3 cm = 1 km)
1) The distance between D and E is km
2) The journey G to E to D is km
3) Travelling from E to G, then to D and on to F is a total of km
4) The trip from F to D and on to E is a total of km
Distances
Try these questions and see how you do
Check each of these answers and tick one of the boxes
| |True | |False |
|A map has a scale of 1 cm = 5km, so 4cm = 20 km | | | |
| | | | |
|On a map with a scale of 1cm = 8 mile, 4 cm represents 2 miles | | | |
| | | | |
|The distance between two towns is measured as 12cm. the map uses a scale of 1cm to 4km so the towns | | | |
|are 3km apart | | | |
| | | | |
|A map uses a scale of 1cm to 8km and the distance between two towns on the map is 5cm. this represents| | | |
|40km | | | |
Use the map below to answer the questions, (Scale 1 cm = 8km)
1) The distance between A and C is km
2) The journey A to B to C is km
3) Travelling from A to C, then to B and on to D is a total of km
Now check yours answers with those on the answer sheet. How did you do
Areas and Volumes
Rules
Area of a rectangle = Length * Width and gives results in e.g. square metres or square cm or square feet etc.
Tip; get used to drawing a diagram and the dimensions
Area = L*W = 3cm * 5cm = 15cm squared, (change to same units where necessary)
E.g. 1m= 100cm
1m= 1000mm
1cm = 10mm
10cm=100mm etc
Class questions
1) A rectangle has sides 15cm and 12cm. Find the area, what would be its perimeter distance?
2) A square has a side of 8cm 5mm, What is its area?
(Note before multiplying, the measurements must be in the same units).
Thus 8cm 5mm= 8.5cm or 85mm
A = L * W = 8.5cm*8.5cm - 72.25cm2
or
85mm *85mm = 7225mm2
Perimeter = 85 + 85 + 85 + 85
Area of rectangles
The area of a rectangle or square = length x width
Find the answers to the following problems:
1) area = 25 m2 length = 5 m width = ?
2) area = 36 m2 length = ? width = 3 m
3) If the area is 153 m2 and the width 9 m calculate the length.
4) 72 sq metres is available for a playground area in the local park. The length must be 12 metres because of the nearby houses. How wide must it be?
5) Jim bought a can of paint which will cover 50 sq metres of fencing. The height of the fence is 2 metres. How many metres length of fence will he be able to paint before running out of paint?
Jim has 70 metres of fencing to paint. How many tins of paint will he need to
buy?
6) 6. A lawn is being laid with new turf. Each roll of turf has an area of 1 sq metre and is 0.5 m wide.
How long is each roll?
The area of a rectangle = length x width
Jeff sees a carpet in the soles that he would like to buy.
area = 24 sq metres width = 4 metres
1) Calculate the length of the carpet.
2) Jeff's room measures 3 metres by 7 metres. Should he buy the carpet? Give reasons for your answer.
3) Extension activity:
If the original cost of the carpet is £95, how much is it in the sale?
Area
(Make sure you use the correct units of measurement for each answer.)
Find the area of the following rectangles.
1
2
3
4
5) A corridor has a rectangular floor of length 10 metres and width 1.5 metres. Find its area.
6) What is the area of a window 0.4 metres wide and 3 metres long?
(Make sure you use the correct units of measurement for each answer.)
Find the length of the following rectangles.
1) Area = 20 cm2, width = 4 cm
2) Area = 16 cm2, width = 4 cm
3) Area = 22m2, width = 10 m
Find the width of the following rectangles.
4) Area = 50 m2, length = 100 m
5) Area = 2.5 cm2, length = 10 cm
Volume
Try these questions and see how you do
Work out the volume of each shape below, being careful to use the correct units.
1)
Volume
2)
Volume
3)
Volume
4)
Volume
5) Work out the volume of a cuboid 16 cm long, 4 cm wide and 10 cm high.
Volume
6) What if the volume of a cube of side length 5cm?
Volume
Comparing fractions and percentages
Fractions to percentages
Write these fractions as percentages.
1) 1/2 =
2) 1/4 =
3) 1/5 =
4) 1/10 =
5) 3/4 =
6) 2/5 =
Percentages to fractions
Write these percentages as fractions, simplifying your answer.
1) 25% =
2) 50% =
3) 75% =
4) 10% =
5) 20% =
6) 30% =
Fractions, Decimals and Percentages
Complete the following table, converting fractions (in lowest terms), decimals and percentages
|Percentage |Fraction (lowest terms) |Decimal |
| |1/5 | |
| | |0.1 |
|60% | | |
| | |0.8 |
| |3/4 | |
|70% | | |
|100% | | |
| |1/2 | |
| | |0.12 |
|17% | | |
| | |0.35 |
Simplify the following fractions
16/50
55/110
15/20
800/900
25/100
48/72
20/80
8/18
Work out the following:-
1) 1/2 of 7 minutes 16 seconds
2) 1/3 of 10 years 3 months
3) 2/7 of 364 days
4) 2/9 of £7.56
5) 3/10 of £81
6) 3/4 of £1000
7) 7/8 of 1000m
8) 3/4 of 84 litres
9) 7/8 of 104 gallons
10) 2/5 of 85km
Finding fraction parts
1) Find:
a) 2/3 of £15
b) 3/4 of £12
c) 4/5 of £10
d) 6/7 of £21
2) When a box of eggs is dropped, 2/3 of the eggs are broken. If the box holds 18 eggs altogether, how many are damaged?
3) Two hundred people take part in a survey. Three-quarters say that they like to go to the cinema. How many people is this?
4) Complete the table below:
|Fraction wanted |Divide by |Multiply by |
|2/3 | | |
|5/7 | | |
| |8 |3 |
| |12 |7 |
|11/ |15 | |
Now try these
Exercise 1
Change these fractions to percentages.
1) 4/5
2) 1/3
3) 3/4
4) 1/5
Now try these
Exercise 2
Change these decimals to percentages.
1) 0.65
2) 0.375
3) 0.89
4) 0.6
5) 0.350
Now try these
Exercise 3
Change these to fractions
1) 25%
2) 30%
3) 75%
4) 4.33 1/3 %
5) 37 1/2 %
Exercise 4
Now convert these to decimals.
1) 63%
2) 4.7%
3) 51.65%
4) 3.1%
5) 7%
Copy out this table, then fill in the gaps
|Fraction |Decimal |Percentage |
|1 |1.0 |100% |
|1/2 |0.5 | |
|3/4 | |75% |
|1/4 |0.25 | |
|1/8 | | |
|1/3 | | |
|2/3 | | |
Exercise 5
1) An apprentice earns £8600 p.a. He is allowed £4000 tax free and pays 20% income tax on the rest. How much income tax does he pay?
2) A company awards a 5% pay rise to all employees. Complete the table below:-
|Name |Current wage |Increase |New wage |
|Paul |£9000 | | |
|Ali |£9600 | | |
|Sue |£9900 | | |
|Pat |£12200 | | |
Find
3) 10% of £3.50
4) 10% of 60cm
5) 10% of 126m
6) 20% of £6.50
7) 6% of 1km
8) 40% of 35m
9) 33 1/3% of 9cm
10) 25% of 52 litres
Percentages
Change these percentages to fractions, in lowest terms:
a) 40%
b) 35%
c) 28%
d) 75%
e) 90%
1) 10% of 45kg
2) 30% of 70
3) 5% of 300
4) 12% of 30
5) 12.5% of 240
Now try these
Exercise 6
1) Increase £200 by 4%
2) Decrease £200 by 4%
3) Increase £420 by 10%
4) Decrease £420 by 10%
Now try these
Exercise 7
What percentage is:
1) 12 out of 48
2) £30 out of £150
3) £200 out of £700
4) 0.5 out of 2.5
5) 1000 out of 8000
-----------------------
23.5
23
24
23.65
23.7
23.6
350cm
6m
1.7 m
375cm
260cm
2.6m
?
?
D
F
8 cm
9 cm
E
6 cm
G
12 cm
B
C
A
D
5 cm
6 cm
2 cm
9 cm
L=5cm
W=3cm
8.5 cm
85mm
6cm
2cm
3cm
3cm
6mm
13mm
8.6m
9m
4 m
8 m
3 cm
3 cm
cube
8 m
15 m
4 cm
6 cm
cube
................
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