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MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text Contents
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text
(b) Ann puts the presents in her suitcase when she packs it to fly home.
What does it weigh now?
(c) If her suitcase now weighs more than 20 kg, there is an extra charge.
She has to pay 15 dollars for every kg or part kg over 20 kg.
How much does Ann have to pay?
1.2 Multiplying and Dividing With DecimalsWhen multiplying or dividing by 10, 100, 1000, etc. the decimal point can simply bemoved to the left or the right, although it should be noted that it is really the digits thatare moving rather than the decimal point. When numbers such as 20, 200 or 300 areinvolved, the numbers can be multiplied by 2 or 3 and then the decimal point can bemoved the correct number of places.
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text
4. A factory produces screws which it sells at 1.2 cents each.
(a) Find the income in cents from the sale of:
(i) 300 screws (ii) 50000 screws (iii) 4000 screws
(b) Convert your answers to (a) from cents to dollars.
(c) $3600 was paid for a batch of screws. How many screws were in thisbatch?
5. A company in the USA made a large profit one year and decided to give a bonus toeach department. The bonus was divided equally among all the staff in eachdepartment.
Department Total Bonus Number of staff
Production $12 487 100
Sales $8 260 20
Delivery $5 350 50
Finance $4 896 40
Find the amount of the bonus that would be paid to staff in each department.
6. A snail moves at a speed of 0.008 miles per hour.
(a) How far would the snail travel in 1.5 hours?
(b) How long would it take the snail to travel:
(i) 40 miles (ii) 0.72 miles?
7. The cost of making a chocolate bar is J$5.27.
(a) What is the cost of producing:
(i) 4000 (ii) 17 000 (iii) 30 000 chocolate bars?
(b) A consultant says that he can reduce the production costs by 40 centsper bar. How much would this save on the production of:
(i) 5000 (ii) 22 000 (iii) 30 000 chocolate bars?
8. A new pop group is trying to produce their first CD.
(a) Their manager finds that it will cost $1.20 to make each CD. If they canafford to spend $1800 on making CDs, how many can they make?
(b) Another CD manufacturer will make the CDs for 90 cents each.How many more can they make at this price?
9. It is established that a truck can carry 64 000 cans of soft drinks. Each can contains0.33 litres of drink.
Find the total volume of the drink carried by the truck.
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text1.2
10. For a major sporting event, a stadium is expected to hold its limit of 70 000spectators.
(a) How much money is taken in ticket sales if the price of the tickets were:
(i) $5 (ii) $8 (iii) $11?
(b) If $432 000 is taken in ticket sales when the ticket price is $6, how manyspectators will not be able to get into the ground?
11. (a) 900 0 6× .
Work out the answer to this sum in your head. Do not use a calculator.
Explain clearly the method you used.
(b) 40 0 8÷ .
Work out the answer to this sum in your head. Do not use a calculator.
Explain clearly the method you used.
12. Write the value of
11 2 0 375 32. .( ) − ÷( )
(a) exactly
(b) to two significant figures.
13. Using a calculator, or otherwise, determine the exact value of
(a) 2 3 4 12 2. .+ (b)0 180 6
0 003..
.−
14. Using a calculator, or otherwise determine the value of 3 483 335
2 3.
..
+ and
write the answer
(a) exactly
(b) correct to one decimal place
(c) correct to one significant figure.
Challenge!
Without moving 6 adjacent numbers of the face of a clock, rearrange the other six so thatthe sum of every pair of adjacent numbers is a prime number.
Information
The sides of the Great Pyramid of Giza in Egypt are about 230.5 m long. Although it wasbuilt thousands of years ago by thousands of slaves, the lengths of its sides vary by nomore than 11.5 cm!
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text
1.3 Fractions and DecimalsSome fractions can be written as decimals with a fixed number of decimal places, forexample:
14
0 25= .
These are called terminating decimals. Others have an infinite number of decimal places,for example:
13
0 333 333= . . . .
Numbers that contain an infinite number of decimal places are usually rounded to aspecified number of significant figures or decimal places.
Remember that significant figures are counted from left to right, starting from the firstnon-zero digit; decimal places are counted after the decimal point.
Worked Example 1
Round each number in the list below to:
(i) 3 significant figures (ii) 3 decimal places.
(a) 4 732.165 (b) 4.736 1 (c) 417.923 5
(d) 0.056 234 (e) 0.004 721
Solution
(a) (i) 4 732 165 4 730. = to 3 significant figures.Note that only the first 3 figures are considered.
(ii) 4 732 165 4 732 165. .= to 3 decimal places. There is no charge as thereare exactly 3 figures behind the decimal point.
(b) (i) 4 736 1 4 74. .= to 3 significant figures. The first three figures areconsidered and the 3 is rounded up to a 4, because it is followed by a 6.
(ii) 4 736 1 4 736. .= to 3 decimal places. The 6 is not rounded up because it isfollowed by a 1.
(c) (i) 417 923 5 418. = to 3 significant figures. The first 3 figures are used andthe 7 is rounded up to 8 because it is followed by a 9.
(ii) 417 923 5 417 924. .= to 3 decimal places. There are three figures behindthe decimal point and the 3 is rounded up to a 4 because it is followed by a 5.
(d) (i) 0 056 234 0 056 2. .= to 3 significant figures. Note that the zeros at the startof this number are not counted.
(ii) 0 056 234 0 056. .= to 3 decimal places.
(e) (i) 0 004 721 0 004 72. .= to 3 significant figures. Note that the zeros in frontof the 4 are not counted.
(ii) 0 004 721 0 005. .= to 3 decimal places. The 4 is rounded up to a 5because it is followed by a 7.
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text
1.5 Operations with Negative NumbersThe following rules for working with negative numbers are important to remember.
Multiplication Division
+ × + ⇒ + + ÷ + ⇒ ++ × − ⇒ − + ÷ − ⇒ −
− × + ⇒ − − ÷ + ⇒ −
− × − ⇒ + − ÷ − ⇒ +
When multiplying and dividing negative numbers, if the signs are the same the result willbe a positive number; if the signs are different the result will be a negative number.
You can see how to use these rules in the following Worked Examples.
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text
1.6 Estimating AnswersIf you do a calculation such as
4 1721 3 84618 21 5 73. .
. .×+
you need to use a calculator to find the answer. This section looks at ways of estimatingthe answers to calculations such as this, and giving answers to a specified accuracy.
Worked Example 1
Give each of the following numbers to (i) 2 significant figures (2 s.f.)
(ii) 1 decimal place (1 d.p.)
(a) 17.47 (b) 0.0784 (c) 4.96
Solution(i) (a) 17.47 ≈ 17 to 2 s.f.
(b) 0.0784 ≈ 0.078 to 2 s.f.
(c) 4.96 ≈ 5.0 to 2 s.f.
(ii) (a) 17.47 ≈ 17.5 to 1 d.p.
(b) 0.0784 ≈ 0.1 to 1 d.p.
(c) 4.96 ≈ 5.0 to 1 d.p.
Worked Example 2
Estimate the answers to each of the following problems.
(a) 18 42 3 76. .× (b) 47 9324 071
..
(c)18 51 11 23
3 0712. .
.+
SolutionEstimates can be obtained by using each number correct to 1 or 2 significant figures (s.f.).
Now, using a calculator, find the answer to each problem in Question 2, givingyour answer to 4 significant figures. In each case compare your answers andestimates.
3. Estimate the answers to each of the following calculations.
(a)6 6 9 5
32 4. .
.×
(b)0 32 8 43
6 21. .
.×
(c)12 8 45 3
17 3. .
.+
(d)33 6 77 9
15 72. .
.+
(e)888 723
38 4+.
(f)560 2 01
29 47+ ..
(g)16 5 3 82
4 162. .
.×
(h)82 4 91 91 04 1 43
. .. .
++
(i)82 6 19 410 024 405
. ..
××
4. When cars leave a factory they are parked in lines until they are delivered.The length of each car is 4.32 m. A line contains 54 cars.
(a) Estimate the length of a line, if there are no gaps between the cars.
(b) Find the length of a line if there are no gaps between the cars.
(c) If there is a gap of 0.57 m between each car, estimate the line length andfind the actual length.
5. A cross-country runner has an average speed of 6 43 1. ms− .
Average speed Distance
Time=⎛
⎝⎞⎠
(a) Estimate and find the distance run in 200 seconds, if he runs at his averagespeed.
(b) Estimate and find, to 3 significant figures, the time it takes him to run1473 m.
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text1.6
6. Drivers at a motor racing circuit complete practice laps in times of 130.21, 131.43and 133.62 seconds. The length of the circuit is 5214 metres.
(a) Estimate the average speed of the drivers. Average speed Distance
Time=⎛
⎝⎞⎠
(b) Find their speeds correct to 2 decimal places.
7. A car travels 12.43 km on 1.12 litres of gasoline.
(a) Estimate and then calculate the distance that the car would travel on 1 litre ofgasoline.
(b) Estimate the distances that the car would travel on
(i) 41.1 litres and (ii) 33.8 litres
of gasoline.
8. A factory in the USA produces 108 CD players every day. The cost of producingthe CD players is made up of $4125 for labour costs and $2685 for parts.
Estimate and then calculate:
(a) the total cost of producing a CD player,
(b) the cost of the parts to make a CD player,
(c) the cost of the labour to make a CD player.
9. Vinyl floor tiles are made so that they are square with sides of length 48 cm.
Estimate and then calculate the number of tiles needed for rooms with sizes:
(a) 6.41 m by 3.28 m (b) 3.84 m by 2.91 m (c) 4.29 m by 4.6 m.
10. (a) Write down the numbers you could use to get an approximate answer to
59 32×
(b) Write down your approximate answer.
(c) Using a calculator find the difference between your approximate answer andthe exact answer.
11. Flour costs J$96.5 per kilogram. Ryan bought 205 kg and shared it equallyamong 14 people. He calculated that each person should pay J$141.30.
Without using a calculator, use a rough estimate to check whether this answer isabout the right size.
You must show all your working.
Challenge!
The growth rate of the human hair varies from person to person. On average, a humanhair grows at a rate of 0.35 mm per day. If the length of a hair is 6 cm, how long will ittake the hair to grow to a length of 26 cm?
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text
1.7 Using Brackets and Memory on aCalculatorBy using the bracket and memory keys on a calculator it is possible to carry out tasksfairly quickly and easily.
Some of the standard memory keys which are found on a calculator are:
Places the current number into the memory, replacing any previous number.
Clears the memory.
Adds the number displayed to the memory.
Recalls the number that is currently in the memory.
Brackets can be used to tell the calculator the order in which to do calculations.
For example, to find:
3 62 4 783 9 1 4. .
. .+−
use
Worked Example 1
Find:
(a)3
3 2 1 8. .+(b) 5 2 3 6
4 7. .
.−⎛
⎝⎞⎠
Solution(a) Use the brackets as shown below
to obtain 0.6.
(b) Use brackets to enclose the top part of the fraction ,as shown below,
MEP Jamaica: STRAND A UNIT 1 Decimals and Fractions: Student Text1.7
13. William uses his calculator to work out
4 2 863 2 0 47
.. .
××
He is told to do this in one sequence, writing down only the answer. He presses thekeys as follows:
This gives him the wrong answer. Explain what is wrong with William's method.
.4 2 ÷6 0 . 43 2 =8 .× × 7
Investigation
In his will, a man left 23 cows to his three children. The eldest child was to have half of theherd, the second child should have one third and the youngest one eighth. The children couldnot decide how to divide up the cows without it being necessary to kill any of them.
A wise man came to the scene. He brought along his only cow and put it with the other 23cows to give a total of 24 cows. He gave half of the 24 cows (12) to the eldest child, one thirdof the 24 cows (8) to the second child and one eighth of the 24 cows to the youngest child. Hethen took his own cow back. What is wrong with this solution?
Investigation
Within 4 consecutive years, Mrs Morton gave birth to four lovely children. Today, x yearslater, Mr and Mrs Morton find out that the product of their four children's ages is 3024.
How old is each child now, assuming that all the children are of different ages?
Investigation
In country X, only 5 cent and 8 cent stamps are available. You have to post letters whichcost 23 cents, 27 cents, 77 cents and $19.51 respectively. Which of these amounts canyou make exactly?
Make a complete list of the amounts between 1 cent and 99 cents which cannot be madeexactly.