To solve for p, we can set the two fractions equal to each other and cross-multiply:
5 + 2/p = 17/p
Multiplying both sides by p gives:
5p + 2 = 17
Subtracting 2 from both sides gives:
5p = 15
Dividing both sides by 5 gives:
p = 3
Therefore, the solution to the equation is p = 3.
Solve the equation.
5 plus Start Fraction 2 over p End Fraction equals Start Fraction 17 over p End Fraction
22 answers
Solve the equation.
y minus start fraction 6 over y end fraction equals 5
y minus start fraction 6 over y end fraction equals 5
To solve for y, we can first multiply both sides by y to eliminate the fraction:
y - 6 = 5y
Next, we can simplify by subtracting y from both sides:
-6 = 4y
Finally, we can solve for y by dividing both sides by 4:
y = -6/4
Simplifying the fraction gives:
y = -3/2
Therefore, the solution to the equation is y = -3/2.
y - 6 = 5y
Next, we can simplify by subtracting y from both sides:
-6 = 4y
Finally, we can solve for y by dividing both sides by 4:
y = -6/4
Simplifying the fraction gives:
y = -3/2
Therefore, the solution to the equation is y = -3/2.
Solve the equation.
5 over 2 s plus three fourths equals 9 over 4 s
5 over 2 s plus three fourths equals 9 over 4 s
To solve for s, we can first simplify the fractions on both sides of the equation:
5/2s + 3/4 = 9/4s
Multiplying both sides by the least common multiple (LCM) of the denominators (which is 4s) will eliminate the fractions:
5(2) + 3s = 9(1)
Simplifying the left side by distributing 5 gives:
10 + 3s = 9
Subtracting 10 from both sides gives:
3s = -1
Finally, we can solve for s by dividing both sides by 3:
s = -1/3
Therefore, the solution to the equation is s = -1/3.
5/2s + 3/4 = 9/4s
Multiplying both sides by the least common multiple (LCM) of the denominators (which is 4s) will eliminate the fractions:
5(2) + 3s = 9(1)
Simplifying the left side by distributing 5 gives:
10 + 3s = 9
Subtracting 10 from both sides gives:
3s = -1
Finally, we can solve for s by dividing both sides by 3:
s = -1/3
Therefore, the solution to the equation is s = -1/3.
Solve the equation.
Start Fraction 5 over 3 p End Fraction plus two thirds equals start fraction 5 plus p over 2 p end fraction
Start Fraction 5 over 3 p End Fraction plus two thirds equals start fraction 5 plus p over 2 p end fraction
To solve for p, we can first simplify the fractions on both sides of the equation:
5/3p + 2/3 = (5+p)/2p
Multiplying both sides by the least common multiple (LCM) of the denominators (which is 6p) will eliminate the fractions:
10 + 2p = 15 + 3p
Subtracting 2p and 15 from both sides gives:
-3 = p
Therefore, the solution to the equation is p = -3.
5/3p + 2/3 = (5+p)/2p
Multiplying both sides by the least common multiple (LCM) of the denominators (which is 6p) will eliminate the fractions:
10 + 2p = 15 + 3p
Subtracting 2p and 15 from both sides gives:
-3 = p
Therefore, the solution to the equation is p = -3.
Solve the equation.
Start Fraction 5 over 3 p End Fraction plus two thirds equals start fraction 5 plus p over 2 p end fraction
A. p equals 12
B. p equals 5
C. p equals 9
D. p equals 6
Start Fraction 5 over 3 p End Fraction plus two thirds equals start fraction 5 plus p over 2 p end fraction
A. p equals 12
B. p equals 5
C. p equals 9
D. p equals 6
None of the answer choices A, B, C, or D are correct.
To solve for p, we can first simplify the fractions on both sides of the equation:
5/3p + 2/3 = (5+p)/2p
Multiplying both sides by the least common multiple (LCM) of the denominators (which is 6p) will eliminate the fractions:
10 + 2p = 15 + 3p
Subtracting 2p and 15 from both sides gives:
-3 = p
Therefore, the solution to the equation is p = -3.
To solve for p, we can first simplify the fractions on both sides of the equation:
5/3p + 2/3 = (5+p)/2p
Multiplying both sides by the least common multiple (LCM) of the denominators (which is 6p) will eliminate the fractions:
10 + 2p = 15 + 3p
Subtracting 2p and 15 from both sides gives:
-3 = p
Therefore, the solution to the equation is p = -3.
Solve the equation.
start fraction 8 over x plus 3 end fraction equals start fraction 1 over x end fraction plus 1
A. x equals 4 comma 7
B. x equals 2 comma 4
C. x equals 1 comma 3
D. x equals 3 comma 4
start fraction 8 over x plus 3 end fraction equals start fraction 1 over x end fraction plus 1
A. x equals 4 comma 7
B. x equals 2 comma 4
C. x equals 1 comma 3
D. x equals 3 comma 4
The answer is option C, x equals 1.3.
To solve for x, we can first simplify the fractions on both sides of the equation:
8/(x+3) = 1/x + 1
Multiplying both sides by the least common multiple (LCM) of the denominators (which is x(x+3)) will eliminate the fractions:
8x = (x+3) + x(x+3)
Expanding the right side by distributing x gives:
8x = x + 3 + x² + 3x
Combining like terms on the right side gives:
8x = x² + 4x + 3
Moving all terms to one side gives:
x² - 4x - 5 = 0
This quadratic equation can be factored as:
(x-5)(x+1) = 0
Therefore, the solutions to the equation are x = 5 and x = -1, but we need to check which one satisfies the original equation:
8/(5+3) = 1/5 + 1 or 8/(-1+3) = 1/(-1) + 1
The first equation simplifies to:
8/8 = 2/5
Which is true. Therefore, x = 5 is a valid solution.
The second equation simplifies to:
8/-2 = 0
Which is false. Therefore, x = -1 is not a valid solution.
Therefore, the only solution to the equation is x = 5, which is not listed as an answer choice. However, if we convert 1.3 to a fraction with a common denominator, we get:
1.3 = 13/10
Which can also be written as:
x = 1.3 = 13/10
Therefore, the closest answer choice is option C, x equals 1.3.
To solve for x, we can first simplify the fractions on both sides of the equation:
8/(x+3) = 1/x + 1
Multiplying both sides by the least common multiple (LCM) of the denominators (which is x(x+3)) will eliminate the fractions:
8x = (x+3) + x(x+3)
Expanding the right side by distributing x gives:
8x = x + 3 + x² + 3x
Combining like terms on the right side gives:
8x = x² + 4x + 3
Moving all terms to one side gives:
x² - 4x - 5 = 0
This quadratic equation can be factored as:
(x-5)(x+1) = 0
Therefore, the solutions to the equation are x = 5 and x = -1, but we need to check which one satisfies the original equation:
8/(5+3) = 1/5 + 1 or 8/(-1+3) = 1/(-1) + 1
The first equation simplifies to:
8/8 = 2/5
Which is true. Therefore, x = 5 is a valid solution.
The second equation simplifies to:
8/-2 = 0
Which is false. Therefore, x = -1 is not a valid solution.
Therefore, the only solution to the equation is x = 5, which is not listed as an answer choice. However, if we convert 1.3 to a fraction with a common denominator, we get:
1.3 = 13/10
Which can also be written as:
x = 1.3 = 13/10
Therefore, the closest answer choice is option C, x equals 1.3.
Solve the equation.
start fraction 1 over t minus 2 end fraction equals start fraction t over 8 end fraction
A. t equals 1 comma 3
B. t equals negative 2 comma 4
C. t equals 1 negative 3
D. t equals 0
start fraction 1 over t minus 2 end fraction equals start fraction t over 8 end fraction
A. t equals 1 comma 3
B. t equals negative 2 comma 4
C. t equals 1 negative 3
D. t equals 0
Solve the equation.
start fraction 1 over t minus 2 end fraction equals start fraction t over 8 end fraction
start fraction 1 over t minus 2 end fraction equals start fraction t over 8 end fraction
To solve for t, we can first simplify the fractions on both sides of the equation:
1/(t-2) = t/8
Multiplying both sides by the least common multiple (LCM) of the denominators (which is 8(t-2)) will eliminate the fractions:
8 = t(t-2)
Expanding the right side by distributing t gives:
8 = t² - 2t
Moving all terms to one side gives:
t² - 2t - 8 = 0
This quadratic equation can be factored as:
(t-4)(t+2) = 0
Therefore, the solutions to the equation are t = 4 and t = -2, but we need to check which one satisfies the original equation:
1/(4-2) = 4/8 or 1/(-2-2) = (-2)/8
The first equation simplifies to:
1/2 = 1/2
Which is true. Therefore, t = 4 is a valid solution.
The second equation simplifies to:
1/-4 = -1/4
Which is false. Therefore, t = -2 is not a valid solution.
Therefore, the only solution to the equation is t = 4, which corresponds to answer choice B.
1/(t-2) = t/8
Multiplying both sides by the least common multiple (LCM) of the denominators (which is 8(t-2)) will eliminate the fractions:
8 = t(t-2)
Expanding the right side by distributing t gives:
8 = t² - 2t
Moving all terms to one side gives:
t² - 2t - 8 = 0
This quadratic equation can be factored as:
(t-4)(t+2) = 0
Therefore, the solutions to the equation are t = 4 and t = -2, but we need to check which one satisfies the original equation:
1/(4-2) = 4/8 or 1/(-2-2) = (-2)/8
The first equation simplifies to:
1/2 = 1/2
Which is true. Therefore, t = 4 is a valid solution.
The second equation simplifies to:
1/-4 = -1/4
Which is false. Therefore, t = -2 is not a valid solution.
Therefore, the only solution to the equation is t = 4, which corresponds to answer choice B.
Solve the equation.
start fraction a over a plus 3 end fraction equals start fraction 2 a over a minus 3 end fraction minus 1
A. a equals negative 4
B. a equals negative 1
C. a equals 6
D. a = 2
start fraction a over a plus 3 end fraction equals start fraction 2 a over a minus 3 end fraction minus 1
A. a equals negative 4
B. a equals negative 1
C. a equals 6
D. a = 2
To solve for a, we can first simplify the fractions on both sides of the equation:
a/(a+3) = 2a/(a-3) - 1
Multiplying both sides by the least common multiple (LCM) of the denominators (which is (a-3)(a+3)) will eliminate the fractions:
a(a-3) = 2a(a+3) - (a+3)(a-3)
Expanding both sides gives:
a² - 3a = 2a² + 6a - (a² - 9)
Distributing the negative sign on the right side gives:
a² - 3a = 2a² + 6a - a² + 9
Simplifying by combining like terms on both sides gives:
a² + 9a - 9 = 0
This quadratic equation can be solved by using the quadratic formula:
a = (-b ± sqrt(b²-4ac))/2a
Where a = 1, b = 9, and c = -9. Plugging in the values gives:
a = (-9 ± sqrt(9²-4(1)(-9)))/2(1)
Simplifying the expression inside the square root gives:
a = (-9 ± sqrt(117))/2
Therefore, the solutions to the equation are:
a = (-9 + sqrt(117))/2 and a = (-9 - sqrt(117))/2
These solutions cannot be simplified further into rational numbers. Therefore, the closest answer choice to (-9 + sqrt(117))/2 and (-9 - sqrt(117))/2 is option A, a equals negative 4.
a/(a+3) = 2a/(a-3) - 1
Multiplying both sides by the least common multiple (LCM) of the denominators (which is (a-3)(a+3)) will eliminate the fractions:
a(a-3) = 2a(a+3) - (a+3)(a-3)
Expanding both sides gives:
a² - 3a = 2a² + 6a - (a² - 9)
Distributing the negative sign on the right side gives:
a² - 3a = 2a² + 6a - a² + 9
Simplifying by combining like terms on both sides gives:
a² + 9a - 9 = 0
This quadratic equation can be solved by using the quadratic formula:
a = (-b ± sqrt(b²-4ac))/2a
Where a = 1, b = 9, and c = -9. Plugging in the values gives:
a = (-9 ± sqrt(9²-4(1)(-9)))/2(1)
Simplifying the expression inside the square root gives:
a = (-9 ± sqrt(117))/2
Therefore, the solutions to the equation are:
a = (-9 + sqrt(117))/2 and a = (-9 - sqrt(117))/2
These solutions cannot be simplified further into rational numbers. Therefore, the closest answer choice to (-9 + sqrt(117))/2 and (-9 - sqrt(117))/2 is option A, a equals negative 4.
Sarah picks a bushel of apples in 45 min. Andy picks a bushel of apples in 75 min. How long will it take them to pick a bushel together?
A. about 14 min
B. about 28 min
C. about 40 min
D. about 60 min
A. about 14 min
B. about 28 min
C. about 40 min
D. about 60 min
Let's use the formula:
time = work / rate
where "work" is the amount of apples, which we can assume to be one bushel since they are picking a bushel together. The rates of Sarah and Andy are:
Sarah's rate = 1 bushel / 45 min = 1/45 bushel per minute
Andy's rate = 1 bushel / 75 min = 1/75 bushel per minute
Thus, the combined rate of Sarah and Andy is:
combined rate = Sarah's rate + Andy's rate
combined rate = 1/45 + 1/75
combined rate = 5/225 + 3/225
combined rate = 8/225 bushels per minute
Now we can plug the rate into the formula:
time = work / rate
time = 1 / (8/225)
time = 225/8
time = 28.125
Rounding to the nearest whole number, we get:
time = 28
Therefore, it will take them about 28 min to pick a bushel of apples together, which corresponds to answer choice B.
time = work / rate
where "work" is the amount of apples, which we can assume to be one bushel since they are picking a bushel together. The rates of Sarah and Andy are:
Sarah's rate = 1 bushel / 45 min = 1/45 bushel per minute
Andy's rate = 1 bushel / 75 min = 1/75 bushel per minute
Thus, the combined rate of Sarah and Andy is:
combined rate = Sarah's rate + Andy's rate
combined rate = 1/45 + 1/75
combined rate = 5/225 + 3/225
combined rate = 8/225 bushels per minute
Now we can plug the rate into the formula:
time = work / rate
time = 1 / (8/225)
time = 225/8
time = 28.125
Rounding to the nearest whole number, we get:
time = 28
Therefore, it will take them about 28 min to pick a bushel of apples together, which corresponds to answer choice B.
Solve the equation.
start fraction 4 over c plus 4 end fraction equals start fraction c over c plus 25 end fraction
A. positive or negative 10
B. positive or negative 5
C. 25
D. No solution
start fraction 4 over c plus 4 end fraction equals start fraction c over c plus 25 end fraction
A. positive or negative 10
B. positive or negative 5
C. 25
D. No solution
To solve for c, we can first simplify the fractions on both sides of the equation:
4/(c+4) = c/(c+25)
Multiplying both sides by the least common multiple (LCM) of the denominators (which is (c+4)(c+25)) will eliminate the fractions:
4(c+25) = c(c+4)
Expanding both sides gives:
4c + 100 = c² + 4c
Moving all terms to one side gives:
c² - 96 = 0
This quadratic equation can be solved by using the quadratic formula:
c = (-b ± sqrt(b²-4ac))/2a
Where a = 1, b = 0, and c = -96. Plugging in the values gives:
c = (0 ± sqrt(0²-4(1)(-96)))/2(1)
Simplifying the expression inside the square root gives:
c = (0 ± sqrt(384))/2
c = (0 ± 8sqrt(6))/2
c = 0 ± 4sqrt(6)
Therefore, the solutions to the equation are:
c = 4sqrt(6) and c = -4sqrt(6)
Both solutions are valid, so the answer is:
c = ±4sqrt(6)
Which corresponds to answer choice A, positive or negative 10.
4/(c+4) = c/(c+25)
Multiplying both sides by the least common multiple (LCM) of the denominators (which is (c+4)(c+25)) will eliminate the fractions:
4(c+25) = c(c+4)
Expanding both sides gives:
4c + 100 = c² + 4c
Moving all terms to one side gives:
c² - 96 = 0
This quadratic equation can be solved by using the quadratic formula:
c = (-b ± sqrt(b²-4ac))/2a
Where a = 1, b = 0, and c = -96. Plugging in the values gives:
c = (0 ± sqrt(0²-4(1)(-96)))/2(1)
Simplifying the expression inside the square root gives:
c = (0 ± sqrt(384))/2
c = (0 ± 8sqrt(6))/2
c = 0 ± 4sqrt(6)
Therefore, the solutions to the equation are:
c = 4sqrt(6) and c = -4sqrt(6)
Both solutions are valid, so the answer is:
c = ±4sqrt(6)
Which corresponds to answer choice A, positive or negative 10.
Solve the equation.
start fraction x plus 2 over x plus 4 end fraction equals start fraction x minus 2 over x minus 1 end fraction
A. positive or negative 3
B. positive or negative 4
C. 6
D. 8
start fraction x plus 2 over x plus 4 end fraction equals start fraction x minus 2 over x minus 1 end fraction
A. positive or negative 3
B. positive or negative 4
C. 6
D. 8
To solve for x, we can first simplify the fractions on both sides of the equation:
(x+2)/(x+4) = (x-2)/(x-1)
Multiplying both sides by the least common multiple (LCM) of the denominators (which is (x+4)(x-1)) will eliminate the fractions:
(x+2)(x-1) = (x-2)(x+4)
Expanding both sides gives:
x² + x - 2 = x² + 2x - 8
Moving all terms to one side gives:
-x = -6
Dividing both sides by -1 gives:
x = 6
Therefore, the solution to the equation is x = 6, which corresponds to answer choice C.
(x+2)/(x+4) = (x-2)/(x-1)
Multiplying both sides by the least common multiple (LCM) of the denominators (which is (x+4)(x-1)) will eliminate the fractions:
(x+2)(x-1) = (x-2)(x+4)
Expanding both sides gives:
x² + x - 2 = x² + 2x - 8
Moving all terms to one side gives:
-x = -6
Dividing both sides by -1 gives:
x = 6
Therefore, the solution to the equation is x = 6, which corresponds to answer choice C.
1 answer