Add.
start fraction 5 c over 2 c plus 7 end fraction plus start fraction c minus 28 over 2 c plus 7 end fraction
A. start fraction 6 c plus 28 over 4 c plus 14 end fraction
B. start fraction 6 c minus 28 over 4 c plus 14 end fraction
C. start fraction 6 c minus 28 over 2 c plus 7 end fraction
D. start fraction 4 c minus 28 over 2 c plus 7 end fraction
Simplifying each fraction by finding a common denominator of 2c + 7, we have:
start fraction 5c(2c-7) over (2c+7)(2c-7) end fraction + start fraction c-28 over 2c+7 end fraction
= start fraction (10c^2 - 35c) / (2c+7)(2c-7) end fraction + start fraction c-28 over 2c+7 end fraction
Combining the two fractions by finding a common denominator of (2c+7)(2c-7), we have:
start fraction (10c^2 - 35c + (c-28)(2c-7)) / (2c+7)(2c-7) end fraction
Simplifying the numerator, we get:
start fraction (10c^2 - 35c + 2c^2 - 49c + 196) / (2c+7)(2c-7) end fraction
= start fraction (12c^2 - 84c + 196) / (2c+7)(2c-7) end fraction
= start fraction 6(c-7) / (2c+7) end fraction
Therefore, the answer is C. start fraction 6 c - 28 over 2 c + 7 end fraction.
Subtract.
start fraction 3 n plus 2 over n plus 4 end fraction minus start fraction n minus 6 over n plus 4 end fraction
A. 2 n plus 8
B. start fraction 2 n minus 4 over n plus 4 end fraction
C. start fraction 2 n minus 8 over n plus 4 end fraction
D. 2
Using the common denominator of n + 4, we have:
start fraction (3n+2) - (n-6) over n+4 end fraction
= start fraction 3n+2 - n + 6 over n+4 end fraction
= start fraction 2n+8 over n+4 end fraction
Therefore, the answer is A. 2n+8.
Find the LCD of the pair of expressions.
one half; Start Fraction 4 over x squared End Fraction
A. 4 x squared
B. 2 x squared
C. 2 x
D. x squared
The prime factorization of the denominator of the fraction 4/x^2 is x^2, so the LCD of the two expressions is x^2.
Therefore, the answer is D. x^2.
Find the LCD of the pair of expressions.
Start Fraction 8 over 5 b End Fraction; Start Fraction 12 over 7 b-cubed c End Fraction
A. 7 b cubed c
B. 12 b cubed c squared
C. 35 b cubed c squared
D. Start Fraction a minus b + 2 over a b c cubed End Fraction
The prime factorization of the denominators are:
- 8/5b has a denominator of 5b.
- 12/7b^3c has a denominator of 7b^3c.
The LCD must include all the factors in both denominators, with the highest exponent of each factor included. Therefore, the LCD is 35b^3c.
Therefore, the answer is C. 35 b cubed c squared.
Find the LCD of the pair of expressions.
start fraction 3 m over m plus n end fraction; start fraction 3 n over m minus n end fraction
A. Left parenthesis m plus n right parenthesis times left parenthesis m minus n right parenthesis
B. 2 m
C. 2 m n
D. left parenthesis m minus n right parenthesis squared
The prime factorization of the denominators is:
- m + n
- m - n
Since these expressions have no common factors, the LCD is the product of the two denominators, which is:
(m + n)(m - n)
Therefore, the answer is A. (m + n) times (m - n).
Add.
start fraction 7 over 3 a end fraction plus 2 fifths
A. start fraction 35 plus 6 a over 3 a plus 5 end fraction
B. start fraction 35 plus 6 a over 15 a end fraction
C. start fraction 7 plus 6 a over 30 a end fraction
D. start fraction 35 plus 6 a over 30 a end fraction
Using the common denominator of 15a, we have:
start fraction (35 + 6a) / 15a end fraction + start fraction 2/5 end fraction
= start fraction (35 + 6a) / 15a end fraction + start fraction 6/15 end fraction
= start fraction (35 + 6a + 2) / 15a end fraction
= start fraction (6a + 37) / 15a end fraction
Therefore, the answer is B. start fraction 35 plus 6 a over 15 a end fraction.
Add.
start fraction a over a plus 3 end fraction plus start fraction a plus 5 over 4 end fraction
A. start fraction 4 a plus 20 over left parenthesis a plus 3 right parenthesis times left parenthesis a plus 5 right parenthesis end fraction
B. start fraction a squared plus 12 a plus 15 over 4 times left parenthesis a plus 3 right parenthesis end fraction
C. start fraction a squared plus 19 a plus 8 over 4 times left parenthesis a plus 3 right parenthesis end fraction
D. start fraction 4 a squared plus 19 a plus 8 over 4 left parenthesis a plus 3 right parenthesis end fraction
Using the common denominator of 4(a+3), we have:
start fraction a/(a+3) + start fraction a+5/4 = start fraction 4a/(4a+12) + start fraction (a+3)(a+5)/(4a+12)
= start fraction 4a + (a+3)(a+5) / 4(a+3)
= start fraction 4a + a^2 + 8a + 15 / 4(a+3)
= start fraction a^2 + 12a + 15 / 4(a+3)
Therefore, the answer is B. start fraction a squared plus 12 a plus 15 over 4 times left parenthesis a plus 3 right parenthesis.
Add.
9 plus start fraction x minus 3 over x plus 2 end fraction
A. start fraction 10 x plus 15 over x plus 2 end fraction
B. start fraction 27 x minus 3 over x plus 2 end fraction
C. start fraction x minus 6 over 9 x plus 18 end fraction
D. start fraction x minus 6 over x plus 2 end fraction
Using the common denominator of x + 2, we have:
9 + start fraction x-3 / x+2 = start fraction 9(x+2) / x+2 + start fraction x-3 / x+2
= start fraction 9x+18 / x+2 + start fraction x-3 / x+2
= start fraction 10x+15 / x+2
Therefore, the answer is A. start fraction 10 x plus 15 over x plus 2 end fraction.
Subtract.
Start Fraction 5 over t squared End Fraction minus Start Fraction 4 over t plus 1 End Fraction
A. start fraction 5 t plus 2 over t squared times left parenthesis t plus 1 right parenthesis end fraction
B. start fraction 5 t plus 1 minus 4 t squared over t squared times left parenthesis t plus 1 right parenthesis end fraction
C. start fraction 1 over t squared times left parenthesis t plus 1 right parenthesis end fraction
D. start fraction 5 plus 5 t minus 4 t squared over t squared times left parenthesis t plus 1 right parenthesis end fraction
Using the common denominator of t^2(t+1), we have:
start fraction 5/t^2 - 4/(t+1) = start fraction 5(t+1) - 4t^2/t^2(t+1)
= start fraction 5t + 5 - 4t^2/t^2(t+1)
Therefore, the answer is D. start fraction 5 plus 5 t minus 4 t squared over t squared times left parenthesis t plus 1 right parenthesis end fraction.
Multiple Choice
What is the difference start fraction 5x minus 2 over 4x end fraction minus start fraction x minus 2 over 4x end fraction ?
A. 1
B. Start Fraction x minus 1 over x End Fraction
C. 0
D. The fraction states 3 over 4.
Combining the fractions with a common denominator of 4x, we have:
start fraction 5x-2/4x - start fraction x-2/4x = start fraction (5x-2)-(x-2) / 4x
= start fraction 4x / 4x
= 1
Therefore, the answer is A. 1.
What is the sum start fraction 1 over 2b end fraction plus start fraction b over 2 end fraction ?
A. start fraction b plus 1 over 2b plus 2 end fraction
B. 2b
C. one-fourth
D. start fraction b superscript 2 baseline plus 1 over 2b end fraction
Using a common denominator of 2b, we have:
start fraction 1/2b + b/2 = start fraction 1}{2b} + start fraction b^2 / 2b
= start fraction 1+b^2 / 2b
Therefore, the answer is D. start fraction b^2 + 1 / 2b end fraction.
What is the sum start fraction 1 over g plus 2 end fraction plus start fraction 3 over g plus 1 end fraction ?
A. start fraction 3 over g plus 3 end fraction
B. start fraction g plus 3 over left parenthesis g plus 1 right parenthesis left parenthesis g plus 2 right parenthesis end fraction
C. start fraction 4g plus 7 over left parenthesis g plus 1 right parenthesis left parenthesis g plus 2 right parenthesis end fraction
D. start fraction 2g plus 3 over left parenthesis g plus 1 right parenthesis left parenthesis g plus 2 right parenthesis end fraction
Using a common denominator of (g+1)(g+2), we have:
start fraction 1/(g+2) + 3/(g+1) = start fraction 1(g+1)/[(g+1)(g+2)] + start fraction 3(g+2)/[(g+1)(g+2)]
= start fraction g+1+3g+6 / (g+1)(g+2)
= start fraction 4g+7 / (g+1)(g+2)
Therefore, the answer is C. start fraction 4g plus 7 over left parenthesis g plus 1 right parenthesis left parenthesis g plus 2 right parenthesis end fraction.
What is the difference start fraction r plus 2 over r plus 4 end fraction minus start fraction 3 over r plus 1 end fraction ?
A. start fraction negative 1 over r plus 3 end fraction
B. start fraction r superscript 2 baseline minus 1 over left parenthesis r plus 1 right parenthesis left parenthesis r plus 4 right parenthesis end fraction
C. start fraction r superscript 2 baseline minus 10 over left parenthesis r plus 1 right parenthesis left parenthesis r plus 4 right parenthesis end fraction
D. start fraction r superscript 2 baseline plus 14 over left parenthesis r plus 1 right parenthesis left parenthesis r plus 4 right parenthesis end fraction