Equivalent Fractions
Equivalent Fractions
Slide: 1
Algebraic Fractions Algebraic fractions contain both numbers and variables.
12a 5
Slide: 2
4pq r–3
2 2x
+1 2 x + 8x – 12
Algebraic Fractions Algebraic fractions contain both numbers and variables.
12a 5
4pq r–3
2 = ( 2x
2 ) + 1 ÷ ( x + 8x – 12)
2 2x
+1 2 x + 8x – 12
Careful! Don’t forget the brackets Slide: 3
Express the following expression in fractional form.
2x + y ÷ (x – 1) – 1 ÷ x – y
Slide: 4
Recall: Equivalent Fractions When both the numerator and denominator of a fraction are multiplied or divided by the same number, the result is a new fraction called an equivalent fraction.
3 5
=
3×2 5×2
=
3×3 5×3
6 9 are equivalent to 3 and 10 15 5 Slide: 5
Equivalent Algebraic Fractions An equivalent fraction can also be obtained by multiplying or dividing the numerator and denominator of a fraction by the same term or expression.
2 3a
5b
=
6a2b 10b2 Slide: 6
2 3a (2b)
5b(2b) is equivalent to
=
2 10b 3a2 5b
Simplifying a Fraction A fraction where the numerator and denominator have no factors in common (other than 1) is said to be a fraction in its lowest (or simplest) terms.
40 45 3 18p 30pq Slide: 7
÷5 ÷5 ÷6p ÷ 6p
8 9 2 3p 5q
8 is the fraction’s 9 simplest form.
3p2 is the fraction’s 5q simplest form.
Simplifying Algebraic Fractions Step 1: Completely factor the numerator and denominator.
HCF = 6x
2 30x
6x – 6x(1 – 5x) = 9x2 + 12x4 3x2(3 + 4x2) HCF = 3x2
Slide: 8
Simplifying Algebraic Fractions Step 2: Identify common factors between the numerator and denominator, and divide the numerator and denominator by the common factors.
6x(1 – 5x) 3x2(3 + 4x2) The numerator and denominator have a common factor of 3x. Slide: 9
÷3x ÷ 3x
2(1 – 5x) = x(3 + 4x2) This is the simplified version of 6x – 30x2 9x2 + 12x4
Completely simplify the following fraction:
10p2qr2 4 2p r
Slide: 10
Completely simplify the following fraction: The numerator and denominator are multinomials. Therefore, we must factor the expressions individually before we can identify common factors between the numerator and denominator.
6ab2 – 12ab 20a2bc – 40a2c
Slide: 11
Completely simplify the following fraction:
4x2 + 16x – 84 4x2 – 36
Slide: 12
Simplifying Algebraic Fractions – Tips Caution! You cannot divide the numerator and denominator by a common term. You can only divide by common factors.
x2
3+ 3 = 2 8+x 8
Slide: 13
3x2 3 = 8x2 8
Simplifying Algebraic Fractions – Tips A difference between two terms and a difference in which the terms are inversed can be divided out by multiplying the fraction by –1.
Notice
–
Slide: 14
–(x – y) = –x + y = y – x
(2x – 1)(x – 3) (3 – x)
= –(2x – 1) = –2x + 1
Completely simplify the following fraction:
2a3 + 12a2 – 3a – 18 3b – 2a2b
Slide: 15
Expanding Fractions When the denominator of a fraction is increased by a factor, the numerator must be increased by the same factor. This is called expanding a fraction.
5 12 2 p 3q Slide: 16
×3 ×3 ×2qr ×2qr
15 36 2 2p qr 2 6q r
15 is an expanded 36 fraction.
2p2qr is an expanded 6q2r fraction.
Expanding Algebraic Fractions Step 1: Identify the multiplying factor by which the denominator increased.
2z 7x
Slide: 17
=
? 3 28x y
Expanding Algebraic Fractions Step 2: Multiply the numerator by the same multiplying factor.
Slide: 18
Expand to solve for ?:
–2 = 2 a + b – 3c
Slide: 19
? 3a(a + b2 – 3c)
Expand to solve for ?:
q = 2p
Slide: 20
? 2pq – 2p
Slide: 1
Algebraic Fractions Algebraic fractions contain both numbers and variables.
12a 5
Slide: 2
4pq r–3
2 2x
+1 2 x + 8x – 12
Algebraic Fractions Algebraic fractions contain both numbers and variables.
12a 5
4pq r–3
2 = ( 2x
2 ) + 1 ÷ ( x + 8x – 12)
2 2x
+1 2 x + 8x – 12
Careful! Don’t forget the brackets Slide: 3
Express the following expression in fractional form.
2x + y ÷ (x – 1) – 1 ÷ x – y
Slide: 4
Recall: Equivalent Fractions When both the numerator and denominator of a fraction are multiplied or divided by the same number, the result is a new fraction called an equivalent fraction.
3 5
=
3×2 5×2
=
3×3 5×3
6 9 are equivalent to 3 and 10 15 5 Slide: 5
Equivalent Algebraic Fractions An equivalent fraction can also be obtained by multiplying or dividing the numerator and denominator of a fraction by the same term or expression.
2 3a
5b
=
6a2b 10b2 Slide: 6
2 3a (2b)
5b(2b) is equivalent to
=
2 10b 3a2 5b
Simplifying a Fraction A fraction where the numerator and denominator have no factors in common (other than 1) is said to be a fraction in its lowest (or simplest) terms.
40 45 3 18p 30pq Slide: 7
÷5 ÷5 ÷6p ÷ 6p
8 9 2 3p 5q
8 is the fraction’s 9 simplest form.
3p2 is the fraction’s 5q simplest form.
Simplifying Algebraic Fractions Step 1: Completely factor the numerator and denominator.
HCF = 6x
2 30x
6x – 6x(1 – 5x) = 9x2 + 12x4 3x2(3 + 4x2) HCF = 3x2
Slide: 8
Simplifying Algebraic Fractions Step 2: Identify common factors between the numerator and denominator, and divide the numerator and denominator by the common factors.
6x(1 – 5x) 3x2(3 + 4x2) The numerator and denominator have a common factor of 3x. Slide: 9
÷3x ÷ 3x
2(1 – 5x) = x(3 + 4x2) This is the simplified version of 6x – 30x2 9x2 + 12x4
Completely simplify the following fraction:
10p2qr2 4 2p r
Slide: 10
Completely simplify the following fraction: The numerator and denominator are multinomials. Therefore, we must factor the expressions individually before we can identify common factors between the numerator and denominator.
6ab2 – 12ab 20a2bc – 40a2c
Slide: 11
Completely simplify the following fraction:
4x2 + 16x – 84 4x2 – 36
Slide: 12
Simplifying Algebraic Fractions – Tips Caution! You cannot divide the numerator and denominator by a common term. You can only divide by common factors.
x2
3+ 3 = 2 8+x 8
Slide: 13
3x2 3 = 8x2 8
Simplifying Algebraic Fractions – Tips A difference between two terms and a difference in which the terms are inversed can be divided out by multiplying the fraction by –1.
Notice
–
Slide: 14
–(x – y) = –x + y = y – x
(2x – 1)(x – 3) (3 – x)
= –(2x – 1) = –2x + 1
Completely simplify the following fraction:
2a3 + 12a2 – 3a – 18 3b – 2a2b
Slide: 15
Expanding Fractions When the denominator of a fraction is increased by a factor, the numerator must be increased by the same factor. This is called expanding a fraction.
5 12 2 p 3q Slide: 16
×3 ×3 ×2qr ×2qr
15 36 2 2p qr 2 6q r
15 is an expanded 36 fraction.
2p2qr is an expanded 6q2r fraction.
Expanding Algebraic Fractions Step 1: Identify the multiplying factor by which the denominator increased.
2z 7x
Slide: 17
=
? 3 28x y
Expanding Algebraic Fractions Step 2: Multiply the numerator by the same multiplying factor.
Slide: 18
Expand to solve for ?:
–2 = 2 a + b – 3c
Slide: 19
? 3a(a + b2 – 3c)
Expand to solve for ?:
q = 2p
Slide: 20
? 2pq – 2p
