2-F
Directed fractions and
decimals
KEY CONCEPTS
Fractions and decimals on a number line
Directed numbers include more than just integers. They can be any values on
a number line, including zero, positive and negative whole numbers, fractions
and decimals.
Fractions and decimals on the negative side of a number line are the mirror
image of the fractions and decimals on the positive side of the number line.
Mirror image
7
–
3
2
5
–
3
4
–
3
2
1
–
3
1
–
3
0
1–
3
Mirror image
2–
3
1
4–
3
5–
3
2
7–
3
−
1.0
−
0.8
−
0.6
−
0.4
−
0.2
0
0.2
0.4
0.6
Operations involving fractions and decimals
The negative sign for a fraction can belong to the fraction as a whole or to
−
either the numerator or the denominator. For example, 75 can be thought of as
−( 5), or 5 .
−7
7
The patterns for addition, subtraction, multiplication and division of integers
hold for directed fractions and decimals.
Addition and subtraction
Multiplication and division
Operation and
following sign are
the same.
+ + or − −
Operation and
following sign are
different.
+ − or − +
Both signs are the
same.
+
× + or − × −
+
÷ + or − ÷ −
Both signs are
different.
−
× + or + × −
−
÷ + or + ÷ 1
Combine to +
Combine to −
Answer is +
Answer is −
Operations with directed fractions
When adding or subtracting directed fractions, combine any operation symbols
and following directed number signs into a single operation.
EXAMPLE 1
Calculate
−2
3
+
−1
2.
WRITE
68
−2
3
+
−1
2
=
1
Write the addition of a negative number as a subtraction.
2
Write all fractions with the same denominator.
=
3
Subtract the fractions and write the answer.
=
Maths XPRESS 8
−2
3
−4
6
−7
6
−
−
1
2
3
6
0.8
1.0
When multiplying or dividing with a negative fraction, either:
perform the calculation as if numbers were all positive, and then determine
the sign of the answer
put the negative sign in either the numerator or the denominator.
EXAMPLE 2
Calculate
−2
7
×
−7
9.
WRITE
1
Move the negative sign into the numerator of the fraction
and cancel the common factor of 7.
2
Multiply the numerators taking note of the positive
and negative signs (−2 × −1 = 2). Then multiply the
denominators (1 × 9 = 9) and write the answer.
−2
7
×
−7
9
=
−2
71
=
−2
×
− 71
9
× −1
1×9
2
9
=
EXAMPLE 3
Calculate
−3
4
÷ −2 12.
WRITE
−3
4
÷ −2 12 =
1
Change mixed numbers to improper fractions.
2
• Change the division symbol to a multiplication symbol
and write the reciprocal of the second fraction.
• Cancel the common factor of 2.
=
3
Multiply the numerators taking note of the positive
and negative signs (−3 × −1 = 3). Then multiply the
denominators (2 × 5 = 10) and write the answer.
=
=
Operations with directed decimals
A number line can be used to determine how to add and subtract decimal
numbers.
When a calculation of an addition or subtraction in the negative half of the
number line is difficult, it is possible to perform the equivalent calculation
in the positive half of the number line to determine the size of the answer.
The answer for the calculation in the negative half will
–4 + –3 = –1
have a sign opposite to the answer for the calculation
in the positive section. For example, it is possible to
calculate −4 + 3 by calculating 4 − 3 as shown at right.
−
4 −3 −2 −1
Since 4 − 3 = 1, the answer to −4 + 3 is −1.
Chapter 2
−3
4
−
3
24
÷
×
−5
2
− 1
2
5
−3
× −1
2×5
3
10
4–3=1
0
1
2
3
4
Positive and negative numbers
69
EXAMPLE 4
Evaluate the following.
a 5.3 − −1.8
b −3.64 − −2.9
c −4.2 − 3.8
WRITE
a 1 Write the subtraction of a negative number as an addition.
2
5.3 − −1.8 = 5.3 + 1.8
15.3
Write the addition as a vertical addition and complete.
+ 1.8
7.1
b 1 Write the subtraction of a negative number as an addition.
2
Use a number line to determine an equivalent calculation.
4
–3.64
+ 2.9
2
3
3.64 – 2.9
1
0
1
2
3
−3.64
− −2.9 = −3.64 + 2.9
3.64
− 2.90
0.74
4
Complete the equivalent calculation (3.64 − 2.9).
3
The answer to the original calculation (−3.64 + 2.9) has the
opposite sign to the equivalent. Write the answer.
c 1 Subtraction from a negative number results in a more
negative number. An equivalent calculation is 4.2 + 3.8.
10
2
3
8
6
4
2
0
2
4
6
8
−
3.64 + 2.9 = −0.74
−
4.2 − 3.8
10
Complete the equivalent calculation.
The answer to the original calculation (−4.2 − 3.8) has the
opposite sign to the equivalent calculation.
4.2 + 3.8 = 8.0
−
4.2 − 3.8 = −8.0
LEARNING EXPERIENCES
Direct competition
Equipment: A4 paper
1 Write down two questions involving directed numbers: one with fractions and the other with
decimals. Share your questions with a classmate and check each other’s answers.
2 Give your problems to the teacher or a volunteer and form groups of four or five. Select one person
to be a scribe. On a sheet of paper, write ‘Round One’ and the numbers 1 to 10.
3 The teacher or volunteer will read out random problems that each group has to solve. Calculators are
not allowed!
4 At the end of the round, correct your answers and get a score out of 10. Continue on with rounds of
questions until you run out of questions or time.
70
Maths XPRESS 8