CSEC Mathematics
How can you be competent in Mathematics? Kerwin Springer
https://www.youtube.com/watch? v=wjWCLTeHynY
https://sites.google.com/a/pcc.edu.jm/csec-mathematics/
SECTION 1 – Number Theory and Computation
Number Theory
What is a number?
A number is an arithmetical value, expressed by a word, symbol, or figure, representing a
Numbers are a fascinating part of our lives. They are much more a part of our everyday life than we might realize and would affect many things you might not realize.
Consider your daily routine.
What would life be without numbers?
How would you be affected if no numbers existed?
Types of Numbers
real numbers, factors, even numbers, prime numbers, complex numbers, whole numbers, irrational numbers, natural numbers,
rational numbers, odd numbers, integers, multiples, composite numbers
Exercise
1. The set of ____________ is another name given to the set of counting numbers. It is represented by the symbol N.
N = {1, 2, 3, 4, 5, …}.
2. The set of ______________ is the set of natural or counting numbers and zero. It is represented by the symbol W.
W = {0, 1, 2, 3, 4, …}.
3. The set of _______________is the set of numbers that is exactly divisible by two. For example, {2, 4, 6, 8, 10, 12, …}.
4. The set of ______________is the set of
numbers which cannot be exactly divided by two. For example, {1, 3, 5, 7, 9, 11, 13, …}.
5. The set of _______________ is the set of numbers which have only two factors, one and itself. For example,
6. The set of ________________ is the set of numbers which have more than two factors. For
example, {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …}.
7. The set of __________ of a number is the set of numbers which can divide another number
without leaving a remainder. For example, the set of factors of 15 is {1, 3, 5, 15} and the set of
factors of 18 is {1, 2, 3, 6, 9, 18}.
Note: Factors can be negative or positive.
8. The set of _____________ of a number is the set of numbers which can be divided by another
number without leaving a remainder. For example, the set of multiples of 3 is
{3, 6, 9, 12, 15, 18, …} and the set of multiples
of 7 is {7, 14, 21, 28, 35, 42, …}.
represented by the symbol Z.
Z = {…, - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, …}.
10. The set of ________________ is the set of numbers which can be written as a fraction. For
example, . It is represented by the
symbol Q.
11. The set of _______________ is the set of numbers that cannot be written as a fraction,
recurring or a terminating number. For example,
and e. It is represented
by the symbol Q1 or I.
12. The set of ____ is the set of both the rational and irrational numbers. It is represented by the
Below is a diagram of the real number system.
We note N W Z Q R and Q1 R.
So R = Q Q1. The Venn diagram representing
the set of real numbers is as follows:
Exercise
Place a tick to state the categories each number belongs.
U = R
N W Z Q
Numbers Numbers Numbers Numbers
219
– 6048
0
π
15.97
2.718
Square Numbers
Square numbers are as follows: 12, 22, 32, 42, 52,
62, 72, 82, 92, 102, 112, 122, …
That is, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 225, …
NOTE: Square roots
Cube Numbers
Cube numbers: 13, 23, 33, 43, 53, 63, 73, 83, 93,
103, 113, 123, … That is,
1, 8, 27, 64, 125, 216, 343, …
NOTE: Cube roots
and so.
Order of Operation
Work out the expression in the bracket first, if there is any
Apply power or index e.g. squaring or cubing the number
Multiplication or division is done before
addition or subtraction, in the order they appear from left to right.
Exercise
Simplify the following without the use of a calculator. Hint: a2 – b2 = (a + b)(a – b)
1. 172 – 122 Ans: 145 2. 142 – 92 Ans: 115
3. 162 - 132 Ans: 87 4. 112 – 102 Ans: 21
5. 82 – 52 Ans: 39 6. 212 - 192 Ans: 80
7. 452 - 432 Ans: 176
Finding Square Roots and Cube Roots of a Number
We can find the square root or cube root of a number by first rewriting the number in
exponent/index form, then simplify.
NOTE: (i) (ii)
(iii) (iv) (v) (ax)y = axy
Simplify each of the following, without using calculator.
a. b. c. d. e.
f. g.
h. i. j. k. l. m.
n. o. p. q.
r. s. t.
ANSWERS: a. 2 b. 3 c. 2 d. 3 e. 5 f. 4
g. 8 h. 10 i. 7 j. 9 k. 5
l. 6 m. 7 n. 16 o. 729 p. 16 q.
1000 r. 25 s. 8 t. 8
Number Systems
The Place Value Chart for the base 10 system
1st dec.pl. 2nd dec.pl. 3rd d.p. 4th d.p. 5th d.p. un dr ed Decimal
point th undr
1
00
,0
00
Face Value, Place Value and Value
The face value of a digit is the actual digit that is identified. For example, the face value of
the digit 7 in 2074.931 is 7.
The place value of a digit is the position of the digit in the place value chart. For example, the place value of the digit 7 in 2074.931 is tens.
The value of a digit is the product of the face and place value of the digit. For example, the
value of the digit 7 in 2074.931 is 7 tens or 70.
The Decimal/Denary System (Base 10)
In counting the number of things we always use
of the group used. Our normal counting system is base ten. The group sizes used are
multiples of ten. Hence, our counting system is called the denary system or the decimal
system.
In the denary system we use the ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit has a place value which is a multiple of ten.
Consider 9734.65810 or 9734.658
Face value 9 7 3 4 . 6 5 8
Place Value 103 102 101 100=1 Decimal
point
10– 1 10– 2 10– 3
Value 9 103=9000 700 30 4 0.6 0.05 0.008
9734.65810 = (9x103)+(7x102)+(3x101)+(4x100)+
(6x10– 1)+(5x10– 2)+(8x10– 3)
In the binary or bicimal system, we count in base two. We use the digits 0 and 1. Each
digit in a number has a place value in terms of
powers of two. The largest digit in base two is 1, whereas the smallest digit is 0.
Consider 11101.01012
Face Value 1 1 1 0 1 . 0 1 0 1
Place Value 24 23 22 21 20=1 Decimal
point
2– 1 2– 2 2– 3 2– 4
Value 1 24 1 23 1 22 0 21 1 1 0 2–1 Base Four Numbers
In the base four system, we count in base four. We use the digits 0, 1, 2 and 3. Each digit
in a number has a place value in terms of powers of four. The largest digit in base four is 3,
whereas the smallest digit is 0.
Consider 1320.13214
Face Value
1 3 2 0 . 1 3 2 1
Place Value
place Value
1320.13214 =
Base Eight Numbers
In the base eight system, we count in base eight. We use the digits 0, 1, 2, 3, 4, 5, 6 and 7. Each digit in a number has a place value in terms of
powers of eight. The largest digit in base eight is 7, whereas the smallest digit is 0.
Consider 5720.34618
Digit 5 7 2 0 . 3 4 6 1
Place Value
83 82 81 80=1 Decimal place
8– 1 8– 2 8– 3 8– 4
Value
5720.34618 =
In the base five system, we count in base five. We use the digits 0, 1, 2, 3, and 4. Each digit
in a number has a place value in terms of powers of five. The largest digit in base five is 4,
whereas the smallest digit is 0.
Consider 1420.34215
Digit 1 4 2 0 . 3 4 2 1
Place Value
53 52 51 50=1 Decimal place
5– 1 5– 2 5– 3 5– 4
Value
1420.34215 =
1a) How many digits are there in a:
i. Base 3 system
ii. Base 6 system
iii. Base 7 system
Copy and complete the table below, for the highlighted digit.
Number Face Value Place Value Value
a) 3465.987 3 Thousand 3000
b) 6.0142 0 tenth 0
c) 11.01112 1 2– 2 1 2– 2
d) 4213.1025 4 53 4 53
1) 15.369
2) 0.2579
3) 347.901
4) 8501.69
5) 23.04125
6) 111011.012
7) 260.75418
Counting In Different Number Systems Exercise
Copy and complete the table below.
Base 10 Base 9 Base 8 Base 7 Bas e 6 Bas e 5 Bas e 4 Base 3 Base 2 Base 1
0 0 0 0 0 0
1 1 1 1 1
4 4 4 4 11
5 5 5 5
6 6 6 6
7 7 7 10
8 8 10
9 10
10 11
11 12
12 13
Converting a Number to Base Ten
When converting a number from its number base to base ten, we use the fact that each place value is a power of its base number.
Exercise
Convert each of the following numbers to the denary system (base 10) and vice-versa.
a. 1011012 b. 110two c. 3210four
f. 1203 g. 5302six
ANSWERS: a. 45 b. 6 c. 228
d. 453 e. 107 f. 15 g. 1190
Converting a Number from Base Ten to Another Base
When converting from a base ten number to
another base, we write down the remainders from bottom to top obtained after dividing by the
indicated base.
Convert each of the following base ten numbers to the number base indicated and vice-versa.
a. 14 to base 3 Ans: 1113
b. 1910 to base 5 Ans: 345
c. 15ten to base 2 Ans: 11112
d. 30 to base 4 Ans: 1324
ANSWERS: a. 1123 b. 345 c. 11112 d. 1324 e. 1136 f. 2539 g. 6208
Adding Binary Numbers
The following rules apply when adding binary numbers:
02 + 02 = 02; 12 + 02 = 12 ; 12 + 12 = 102;
102 + 12 = 112; 112 + 12 = 1002
For example, 1112 + 1012
1 1 12
+1 0 12
11 0 02
Exercise
Add the following binary (base two) numbers. a. 11012 + 1112 ANS: 101002
b. 101012 + 10112 ANS: 1000002
c. 11112 + 11112 ANS: 111102
+ 0 1
0 0 1
d. 100112 + 110112 ANS: 1011102
e. 1012 + 1112 + 1102 ANS: 100102
f. 10102 + 10112 + 11012 ANS: 1000102
g. 1011012 + 110112 + 110012 + 1112 ANS: 11010002
Subtracting Binary Numbers
The following rules apply when subtracting binary numbers:
02 - 02 = 02; 12 - 02 = 12 ; 12 - 12 = 02; 102
– 12 = 12; 112 – 12 = 102; 1002 – 12 = 112
For example, 1112 - 1012
1 1 12
-1 0 12
1 02 Exercise
Compute the following.
a. 110112 - 10102 ANS: 100012
c. 1010112 - 111112 ANS: 11002
d. 100102 - 11112 ANS: 112
e. 100002 – 10102 ANS: 1102
f. 10101012 - 1101102 ANS: 111112
Converting from Base 5 to Base 10
1. Convert each of the following numbers to base 10.
a. 32045 ANS: 429
b. 4105 ANS: 105
c. 13045 ANS: 204
d. 23405 ANS: 345
2. Convert each of the following to base 5. a. 429 ANS: 32045
b. 105 ANS: 4105
c. 269 ANS: 20345
Adding Quinary (Base Five) Numbers
The following rules apply when adding base five numbers:
+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 10
2 2 3 4 10 11
3 3 4 10 11 12
4 4 10 11 12 13
For example, 4325 + 1045
4 3 25
+ 1 0 45
1 0 4 15
Exercise
a) 435 + 345 ANS: 1325
b) 43215 + 34125 ANS: 132335
c) 21345 + 10325 ANS: 32215
d) 401235 + 234105 ANS: 1140335
e) 10245 + 13425 + 40145 ANS: 114405
f) 2415 + 1035 + 4105 + 2345 ANS: 20435
g) 402135 + 24135 + 14025 ANS: 1000335 Subtracting Quinary (Base Five) Numbers
For example, 34125 - 2035
3 4 1 25
- 2 0 35
3 2 0 45
Exercise
Subtract the following base five numbers.
a) 3215 – 425 ANS: 2245
b) 41325 – 34325 ANS: 2005
d) 40125 – 31425 ANS: 3205
e) 102435 – 43125 ANS: 4315
f) 3001425 – 132445 ANS: 2313435
g) 124305 – 42035 ANS: 32225 Exercise
Compute the following.
a. 12304 + 23124 ANS: 102024
b. 30124 – 21304 ANS: 2224
c. 1256 + 3026 + 1306 ANS: 10016
d. 26347 – 15267 ANS: 11057
e. 3218 + 20738 + 1368 ANS: 25528
f. 10223 – 2213 ANS: 1013
g. 120324 + 213014 ANS: 333334
Note: We must ensure that both quantities have the same unit before we express one
quantity as a fraction of another.
Exercise
Express the first quantity as a fraction of the second quantity.
1. 20 cents, $3.00 (100 cents = $1)
2. 5 days, 4 weeks
3. 9 months, 5 years
4. 17 cm, 6 m (1 m = 100 cm)
5. 25 kg, 8 tonnes (1 tonne = 1000 kg)
Sequences of Numbers – Page 28 R. Toolsie’s, textbook
number in the sequence is called a term and is given a value according to its position.
Exercise
Identify the rule for each of the following sequence and fill in the blanks.
a. 3, 15, 75, ____
b. 1, 3, 2, 4, 3, ____
c. 7, 6, 8, 7, 9, ____
d. 1, 4, 9, ____, 25
e. 3, 12, 48, ___, ____, 3072
f. - 9, - 6, - 3, 0, 3, _____
g. - 8, - 4, - 2, - 1, ____
h. 5, 6, 9, 14, 21, ____
i. 8, 10, 14, ____, 28, ____
j. 0, 1, 1, 2, 3, ____, 8, 13, ____
k. 0, 1, 8, 27, ___, 125, ____
The commutative law for an arithmetic operation deals with the order in which the
operation is performed, does not affect the result.
Thus: 2 + 6 + 9 = 9 + 2 + 6 = 17 and
2 3 5 = 5 2 3 = 30.
Hence, addition of numbers and the
multiplication of numbers are both commutative.
Now 7 – 2 2 – 7, that is, 5 – 5.
Also, 8 2 2 8, that is 4 . Hence,
subtraction of numbers and division of numbers are both non-commutative.
Associative Law
The associative law for an arithmetic operation deals with grouping the numbers and
2 4 5 = (2 4) 5 = 2 (4 5) = 40. Hence, the addition of numbers and the multiplication
of numbers are both associative.
Now 9 – 5 – 2 = (9 – 5) – 2 9 – (5 – 2),
that is 2 = 2 6 and
8 4 2= (8 4) 2 8 (4 2), that is 1 4. Hence, the subtraction of numbers and the division of numbers are both non-associative.
Distributive Law
The distributive law for an arithmetic operation deals with the multiplication of
numbers in brackets.
3 (4 + 7) = 3 4 + 43 7 = 12 + 21 = 33 and
4 (8 – 3) = 4 8 – 4 3 = 32 – 12 = 20. Hence, multiplication is distributive with
Answer the following.
1. 5×6 + 5×3 is the same as:
A. 5×6
B. 6+3
C. 5×9
D. 5+9
2. 7×5 – 7×2 is the same as:
A. 7×3
B. 5 – 2
C. 7 – 3
D. 5×7
3. 12×8 – 12×3 is the same as:
A. 12×3
B. 12×5
D. 12 – 5
Identity Element for Addition
The identity for an operation leaves the original number unchanged under the operation.
If zero is added to any number, then the sum is the original number. Thus:
i. 4 + 0 = 4 ii. 0 + 3 = 3 iii. – 4 + 0 = – 4 iv. 0 + (– 3) = – 3
Zero is the identity element for the addition of numbers.
Identity Element for Multiplication
If any number is multiplied by 1, then the product is the original number. Thus:
One is the identity element for the multiplication of numbers.
Inverse for Numbers under Addition
The inverse of a number for a given operation combines with the number under the operation to give the identity. Thus:
The inverse of 5 under addition is – 5, since 5 + ( – 5) = 0 (identity).
The inverse of – 3 under addition is 3, since – 3 + 3 = 0 (identity).
Inverse for Numbers under Multiplication
The definition for the inverse was stated above. Thus:
The inverse of 6 under multiplication is , since
The inverse of – 7 under multiplication is ,
since – 7 = 1 (identity)
Multiplication by Zero
If any number is multiplied by zero, the product is always zero. Thus:
i. 8 0 = 0 ii. 0 7 = 0
iii. – 3 0 = 0 iv. 0 ( – 1) = 0
Division by Zero
If any number is divided by zero, the result is infinity. Thus:
i. = ii.
The following table gives the Divisibility Rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.
Divisibilty Rules
3 If the sum of the digits is a multiple of 3 4 If the last two digits is a multiple of 4 5 If the last digit is 0 or 5
6 If the number is a multiple of 2 and 3
7 Cross off last digit, double it and subtract. Repeat if you want. If new number is a multiple of 7, the original number is a multiple of 7
8 If the last 3 digits is a multiple of 8
9 If the sum of the digits is a multiple of 9 10 If the last digit is 0
11 Subtract the last digit from the number formed by the remaining digits. If new number is a multiple of 11, the original number is a multiple of 11
12 If the number is divisible by 3 and 4
Highest Common Factor (H.C.F.)
The H.C.F. of a set of numbers is the highest number which can divide each of the set of numbers without leaving a remainder.
Exercise
Find the H.C.F. for the following set of numbers.
1. 56, 28, 40 ANS: 4
2. 30, 45, 60 ANS: 15
4. 54, 192, 96 ANS: 6
5. 240, 160, 340 ANS: 20
Lowest Common Multiple (L.C.M.)
The L.C.M. of a set of numbers is the lowest number for which each of the set of numbers can go into without leaving a remainder.
Exercise
Find the L.C.M. for the following set of numbers.
1. 2, 5, 6 ANS: 30
2. 4, 8, 12 ANS: 24
3. 5, 9, 6 ANS: 90
4. 7, 8, 14 ANS: 56
5. 3, 7, 9 ANS: 63
Arranging Fractions in Ascending or Descending order
Note: Express each set of fractions with a common denominator. Then compare
Express each fraction as a percentage. Then compare for the order of size.
Exercise
Arrange the following fractions in ascending order (smallest to the largest).
1. ANS:
2. ANS:
3. ANS:
4. ANS:
5. ANS:
Arrange the following fractions in descending order.
1. ANS:
2. ANS:
4. ANS:
5. ANS:
Computation Approximation
An approximation is a stated value of a number that it is close to, but not equal to.
The Place Value Chart
1st d.p. 2nd d.p. 3rd d.p. 4th d.p. 5th d.p. T ho us an ds H un dr ed s T en s U ni ts /O ne s Decima l point T en th H un dr ed th T ho us an dt h T en T ho us an dt h H un dr ed T ho us an dt h
Rounding off to the nearest whole number
number part and leave out the decimal part. However, if the digit value in the first decimal place is less than 5, we do not add 1 but leave out the decimal part.
Example
Write each of the following numbers correct to the nearest whole number.
a. 174.573 Ans. 175
b. 2.999 Ans. 3
c. 25.47 Ans. 25
d. 0.068 Ans. 0
1. 348.379
2. 70.817
3. 19.635
4. 28.15
5. 6999.749
When correcting a number to the nearest ten, we examine the digit value of the units/ones place. If the digit value in the units place is 5 or more, we add 1to the digit value of the tens and leave out the decimal part. However, if the digit value in the units place is less than 5, we do not add 1 but just leave out the decimal part and place zero at the unit place as a place holder.
Example
Write each of the following numbers correct to the nearest tens.
a. 25.36 Ans . 30
b. 12.958 Ans. 10
c. 348.5036 Ans. 350
d. 1,292.36 Ans. 1290
1. 3,456.214
3. 72.369
4. 1699.852
5. 102.598
6. 4.925
7. 7.673
Approximation – Decimal Places
When rounding off a number to the specified number of decimal places, we look at the digit value that follows, reading from left to right. If the digit value that follows is 5 or more, we add 1to the specified decimal digit, keep the whole number part and leave out the decimal part that comes after it. However, if the digit value of the specified decimal place is less than 5, we do not add 1 but just leave out the decimal part that
follows after it and keep the whole number part.
Express each of the following numbers to the number of decimal places indicated in the
bracket.
a. 6.07 (1 d.p.) Ans. 6.1
b. 124.06981 (3 d.p.) Ans. 124.070
c. 0.021458 (2 d.p.) Ans 0.02
1. 17.823 (1 d.p.)
2. 6987.245601 (3 d.p.)
3. 180.50361 (2 d.p.)
4. 8.1973 (1 dec. pl.)
5. 0.09876 (2 dec. pl.)
Approximation – Significant Figures
When rounding off a number correct to a specified number of significant figures, we
the specified number digit value. However, if the digit value that comes after the specified number of significant figure is less than 5, we do not add 1.
NOTE: The first significant figure cannot be zero. The first significant figure of a number is the first non-zero (not zero) digit that occurs in the number, reading from left to right.
Example
Express the number 195.8024 correct to the number of significant figure(s) stated.
a. 6 s.f. Ans. 195.802
b. 5 s.f. Ans. 195.80
c. 4 s.f. Ans. 195.8
d. 3 s.f. Ans. 196
e. 2 s.f. Ans. 200
Express the number 0.00206398457 correct to the number of significant figure(s) stated.
a. 6 s.f. Ans. 0.00206398
b. 5 s.f. Ans. 0.0020640
c. 4 s.f. Ans. 0.002064
d. 3 s.f. Ans. 0.00206
e. 2 s.f. Ans. 0.0021
f. 1 s.f. Ans. 0.002
Standard Form or Scientific Notation
A number which is written in the form a x 10n,
where 1 ≤ a < 10 and n Z is said to be written in standard form or scientific notation.
We move the decimal point in the given number to obtain a number between 0 and 10.
n is positive if the given number is greater than 10 and negative if the given number is between 0 and 1. Otherwise, n = 0.
Example
Express each of the following numbers in standard form (scientific notation).
a. 841902 Ans. 8.41902 x 105
b. 0.00047935 Ans. 4.7935 x 10 – 4
c. 7495 (correct to 1 d.p.) Ans 7.5 x 103
d. 0.057849 (correct to 2 d.p.) Ans.5.78 x 10 – 2
Activity
1. Calculate 2.01 0.015, giving your answer:
a. exactly
b. correct to 3 sig. fig.
c. correct to 2 dec. pl.
e. to the nearest whole number
2. Calculate 47.021 3.6, giving your answer:
a. exactly
b. correct to 2 sig. fig.
c. 3 dec. pl.
d. nearest hundreds
e. in standard form
3. Compute 1718.052 67.8, giving your answer:
a. exactly
b. correct to 3 sig. fig.
c. 1 dec. pl.
d. nearest tens
4. Evaluate 0.0074375 1.7, giving your answer:
a. exactly
b. 4 dec. pl.
c. nearest thousandth
d. 3 sig. fig.
Computation – Fractions and Decimals Fractions
A fraction is part of a whole. The whole is
represented as 100%. A whole is considered as a quantity when the numerator and the denominator are the same. In a fraction, the number at the top is called the numerator and the number at the bottom is called the denominator. The line between the number at the top and the number at the bottom is called the ‘fraction bar’ or vinculum.
The fraction above represents 3 equal parts out of a total of 7 equal parts. The whole would be 7 equal
numerator
denominator
parts out of a total of 7 equal parts which equates to
one. That is, .
A proper fraction is a rational number whose numerator is less than its denominator. For
example, .
An improper fraction is a rational number whose numerator is greater than its denominator. For
example, .
A mixed number consists of the sum of a whole number and a rational number. For example,
.
Note: Activity
a. b. c. d. e.
f. g.
Exercise
Express each of the following improper fractions as a mixed number.
Changing from a Mixed Number to an Improper Fraction
To change a mixed number into an improper fraction, we multiply the whole number by the
denominator and add the numerator, then write the result over the denominator.
Exercise
Express each of the following as an improper fraction.
a. b. c. d.
Equivalent Fractions
reduced to their lowest term. We can make
equivalent fractions by multiplying or dividing the numerator and the denominator by the same value.
Exercise
Determine the equivalent fraction for each of the following fractions with the denominator or
numerator indicated.
a. , denominator 36
b. , numerator 21
d. , numerator 8
e. , numerator 45
f. , denominator 18
g. , numerator 5.
Enrichment
Reduce each of the following fractions to its simplest form.
a. b. c. d. e.
f. g.
Order of Operation
Work out the expression in the bracket first,
Apply power or index e.g. squaring or cubing the number
Multiplication or division is done before addition or subtraction, in the order they appear from left to right.
In the case of a fraction work out the numerator first then the denominator accordingly.
Performing Operations on Fractions and Decimals
Addition and Subtraction of Fractions
Note: We can only add or subtract the numerator of fractions which have a common denominator.
For example,
Note: is the same as .
Multiplication and Division of Fractions
Note: When multiplying or dividing fractions, we do not find a common denominator.
Multiplication of Fractions
When multiplying fractions, we can cancel a
numerator and a denominator if possible. Then we multiply numerator by numerator all over, the
denominator multiplied by the other denominator.
For example, .
Division of Fractions
When dividing by a fraction we multiply by the dividend reciprocal of the divisor. The divisor is the fraction immediately to the right of the division
sign. For example,
1. JANUARY 2016 – Ques. 1 a, b
a)
ANSWER: 2.88
b)
ANSWER:
b)
3.
ANSWER: OR
4.
May 2015
5.
May 2014
6.
Jan. 2014
Answer: 2.40 (2 dec. pl.)
7.
Jan. 2012
8.
ANSWER: OR
Jan. 2011
9.
May 2009
10.
May 2008
11.MAY 2007
ANSWER: i) 86.65
ii)
ANSWER: i)
ANSWER: i) 1.873
ANSWER: a) 8.89
b) i. 40 ii. 720 iii.
12.May 2006
CSEC Mathematics
Fractions and Decimals Worksheet
1. Using a calculator or otherwise, calculate the exact value of:
a. (2.67 4.1) – 1.32. (Jan 2013) Ans: 9.257
b. 5.25 0.015 Ans: 350
c. Ans: 2.55
d. 3.142 x 2.2362 (May 2014) Ans: 15.709
e. ANSWER:
f. (May 2013)
g. (12.8)2 – (30 0.375) (Jan 2015)
2. Give your answer as a fraction in its lowest term:
(May 2012) ANSWER:
3. Evaluate: May 2015
May 2015
a. ANSWER:
b. 3.96 x 0.25 – (May 2011)
c. (Jan 2011)
5. Write your answer as a decimal to 2 significant figures:
a. (June 1989)
ANSWER:
b. (June 1992) ANSWER:
c. (Jan 1990) ANSWER:
6. Calculate the exact value of:
a. (Jun 2005) ANSWER:
b. 0.03 x 10.3
7. Find the exact value and write your answer to 1 decimal place:
a. 2.55 x 6.3 – (Jun 1987)
8. Give your answer to 2 significant figures:
a. (Jun 1990)
b. (Jun 1998)
9. Determine the exact value of:
a. ANSWER:
b. 2.52 – (give answer for b to 2 significant
figures) (May 2010) Ans: 6.1
10. Simplify:
ANSWER: 1.7 (May 2015).
11. Farmer Dan used of his land for oranges,
for apples, for mangoes and the remainder for
a. Calculate the total fraction of the land used for all fruits except grapes. ANSWER: OR
b. What fraction of the land was used for grapes?
ANSWER: OR
12. Mark spends of his monthly income on
housing. Of the remainder, he spends on food
and saves what is left.
a. Calculate the fraction of his monthly income
spent on food. ANSWER:
b. Calculate the fraction of his monthly income that he saved. ANSWER:
Arithmetic Operations on Numbers Order of Operation (reminder)
Work out the expression in the bracket, if
Apply powers or index e.g. squaring or cubing the number
Multiplication or division is done before addition or subtraction, in the order they appear from left to right.
In the case of a fraction work out the numerator first then the denominator.
Exercise
Simplify the following, without the use of a calculator.
1. 24 8 4 2. 24 8 4 3. 5 + 32 4. 28 – 12 4
5. 5 + 3 2
6. (3 2 – 1) + (44 11 – 7) 7. (7 – 4) 9 (8 2 3) 8. 5.32 0.46
13. 45 – 2(4 + 2 3) 14. 10 – (12 – 6 3) 15. 25 – 5 + 16 4 2
Expressing a Decimal Number as a Fraction Exercise
Express each of the following as a fraction in its lowest state.
1) 0.45 2) 0.125 3) 0.375
4) 6.25 5) 12.75 6) 5.02 7) 18.625
Percentage
A percentage is a number or rate expressed out of 100.
For example, 17% is .
Conversion Among Fractions, Decimals and Percentage
Exercise
Copy and complete the table below. Show all working below the table.
Fractions Decimal Percentage
2. 0.125
3. 2.5%
4. 0.65
5. 45%
6.
7. 0.8
8. 25%
9.
10. 0.32
Exercise
Calculate the following. 1. What is 30% of $600?
2. What is 0.45% of $500? Ans: $2.25
3. 25% of a certain volume is 60 cm3.
Calculate the total volume. Ans: 240 cm3
5. Express 17 as a percentage of 60, exactly. 6. There are 530 students in a school and 30% are footballers.
a. What percentage of the school are not footballers?
b. How many students are not footballers? 7. A mathematics book has 360 pages, of which 50% are on Algebra, 20% on Geometry and the remainder on Arithmetic.
a. What percentage of the book is on Arithmetic?
b. How many pages of Arithmetic are
there in the book? Ans: 108 pages
8. The price of a car that cost $27,000 last year increased by 12.5% this year. What is
the present cost of the car? Ans: $30,375
9. A concert is attended by 2500 people. If 47% are adult females and 32% are adult males, how many children attended the concert?
Ans 525 children
10. Express 45 as a percentage of 25. 11. What percentage of 20 is 17?
Finding the square root of a number
Note: , ,
Exercise
Simplify the following, without using a calculator.
1. 2. 3. 4.
5. 6. Ans: 15/4
Ratio
A ratio refers to a comparison between
measures of the same quantity carried out by division.
Exercise
Answer the following.
1. The sum of $25,000 was divided among two consultants in the ratio 2:3. What is the amount of the smaller share?
Answer $10,000
2. A sum of money is divided among two
friends in the ratio 4:11. If the smaller amount is $420, find the larger amount. Answer
$1,155
3. Two lengths are in the ratio 7:8. If the first length is 273m, what is the second length?
Answer 312 m
4. Natasha and Tricia shared a sum of money in the ratio 5:3 respectively. If Tricia’s share was $126.75, calculate:
a) Natasha’s share Answer $211.25[]
b) the total sum of money shared.
5. A sum of money was shared among
Albert, Bruce and Christine in the ratio 3:7:10 respectively. If Bruce received $660 more
than Albert, determine the sum of money
shared. Answer $3,300
6. A sum of money is divided among Yuri, Anna and Maria in the ratio 4:7:9
respectively. If Anna’s share amounts to $1295, calculate:
a) the total sum of money shared
Answer $3700
b) Yuri’s share Answer $740
c) the percentage of the total amount that
Maria received. Answer 45%
7. A sum of money was shared among three daughters, Ann, Beryl and Candy, in the ratio 2:5:8 respectively. If Ann received $510 less than Candy, evaluate the sum of money
shared. Answer $1,275
8. The sum of $3500 is divided among
Adrian received $850 and James received the remainder. Calculate:
a. Sean’s share Answer $1,750
b. James share Answer $900
c. The ratio in which the $3500 was divided among the three persons. ANS: 17:35:18
9. John, Peter and Mary shared a sum of money in the ratio 2:4:9, respectively. Mary and Peter together received $780. How much money in all was shared? Answer $900
10. The sum of $2,040 was shared among Akeme, Shawna and Kissis. Kissis received $720 more than Akeme. Akeme received $360. Determine:
a) Shawna’s amount Answer $600
b) the ratio into which $2,040 was divided among the three persons.
Answer 9:3:5
(Practice Questions, Ex. 3r Page 81)
Direct Proportion – Page 82, R. Toolsie’s textbook
Two quantities are said to be in direct
proportion, they increase or decrease using a constant multiplier. That is, if one quantity is doubled, then the other is doubled also. If we halve one quantity the other is also halved. For example, if the cost of 2 gleaners is $300, then the cost of 4 gleaners would be $600.
Exercise
Answer the following.
1. The cost of 26 articles is $214.50. Calculate the cost of:
a. one article Ans: $8.25
b. 15 articles Ans: $123.75
2. Eggs cost $5.40 per dozen. What is the
3. A train travels 240 km in 42 hours. How
long would it take to complete 350 km?
Ans: 61.25 hours
4. A 5 kg bag of peas cost $17.90. Calculate
the cost of a 9 kg bag of peas at the same rate.
Ans: $32.22
5. A car travels 240 km on 20 litres of petrol.
How many litres of petrol is needed to travel
600 km? Ans: 50 litres
6. The cost of 8 sweets is $320. What is the
cost for 5 sweets? Ans: $200
7. It cost $112 to cut a lawn of area 56 m2.
What amount would it cost to cut a lawn of
Inverse Proportion – Page 87, R. Toolsie’s textbook
One quantity is said to be inversely
proportional to another quantity, if when the first quantity is doubled, the second quantity is halved. And if the first quantity is halved the second quantity is doubled. For example, if two men can weed a compound in 6 days, then 4 men working at the same rate can
weed the compound in 3 days.
Exercise
Answer the following.
1. If 12 men can sew 180 shirts in 5 days, how long will it take 15 men to sew the 180
shirts? Ans: 4 days
2. Twelve men produce 700 watches in 9 working days. How long would it take 18 men to produce the 700 watches?
3. A field of grass feeds 28 cows for 6 days. How many days would the same field feed 21
cows? Ans: 8 days
4. If 9 women can sew 375 dresses in 8 weeks, calculate the time it would take 4 women to perform the same task.
Ans: 18 weeks
5. Nine taps fill a water tank in 3 hours. How many hours would it take to fill the tank if
only three taps are working? Ans: 9 hours
6. A rice farmer employs 15 men to harvest his crop. The men took 12 days to do the job. If he employed 9 men, how many days would it have taken them? Ans: 20 days
7. If 4 men can paint a house in 12 days, how long would it take 6 men to complete the
The Metric System
The metric system consist of basic quantities such as length, mass, time, temperature with corresponding basic units, metre (m), gram (g), seconds(sec), degree Celsius (0C)
respectively.
Conversion from One Unit to Another
Prefix Symbol Multiplication
factor kilo hecto deca deci centi milli k h da d c m 1000 100 10 0.1 0.01 0.001
Length: km, hm, dam, m, dm, cm, mm (descending order of units)
Mass: kg, hg, dag, g, dg, cg, mg (descending order of units)
1 kg = 2.21 pounds 1 tonne = 1000 kg
Note: When changing from a larger unit to a smaller unit we multiply by the respective power of ten.
However, when changing from a smaller unit to a larger unit we divide by the respective power of ten.
Exercise
Write the correct value on the line provided. 1) 34.56 cm = ______m Answer 0.3456 2) 8.107 mm = _____dm
Answer 0.08107
3) 4.72 kg = ______ hg Answer 47.2
4) 345.09 m = ______ km Answer 0.34509
5) 4.5 litre = ______ cm3 Answer 4,500
6) 670.2 cm3 = _____ litres Answer 0.6702
7) 80.321 g = _____ mg Answer 80,321
9) 9.3 litres = _____ cm3 Answer 9,300
10) 45.981 mm = _____ m Answer 0.045981
[Practice question – Page 107 Exercise 4a, R. Toolsie textbook]
Change from 24 –hour to 12 –hour system and vice-versa (Page 154, R. Toolsie)
Exercise
1. Copy and complete the table below.
12-hour system 24-hour system
a. 3:00 am b. 5:00 pm c.
d.
e. 11:40 pm
2. Calculate the length of time in hours and minutes for the following pairs of time
extracted from an airline time schedule. Time of
Departure
Time of Arrival
a. 3:39 a.m. 6:43 a.m.
b. 7:38 a.m. 11:18 a.m.
c. 12:15 p.m. 2:30 p.m.
d. 1:15 p.m. 3:12 p.m.
e. 10:45 a.m. 7:32 p.m.
f. 9:12 p.m. 6:45 a.m. (next day)
g. 8:45 p.m. 7:30 p.m. (next day)
