Please, will someone check my answers and help me solve the other questions?
1. It takes Dustin 2 hours to shovel the snow from his driveway and sidewalk. It takes his sister 3 hours to shovel the same area. How long will it take them to shovel the walk if they work together?
A: Working together, Dustin and his sister can shovel the walk in 1 1/5 hours, or 1 hour and 12 minutes?
2. A chemistry student needs to make a solution that is 70% water and 30% hydrochloric acid. The student's current mixture of 300 ml is 60% water and 40% hydrochloric acid. How much water must the student add to achieve his desired solution?
A: ?
Solve. Check your answer.
3. ((d + 2)/(d + 8)) = ((-6)/(d + 8))
A: ?
4. 4/n^2 = 7/n + 2
A: n = -4 or n = 1/2?
5. 2/t + 4/3t = 4/t + 2
A: t = 10?
6. ((x -6)/(x^2-6)) = ((-4)/(x-4))
A: ?
let the amount to be added by x ml
so we have 300+x ml of the new mixture
.3(300+x) = .4(300)
3(300+x) = 4(300)
900 + 3x = 1200
3x = 300
x = 100
Add 100 ml of water
check:
Do we have the correct amount of acid?
in original: .4(300) = 1200 ml
in new: .3(400) = 120 ml -----> correct
what about the concentrtion of water?
in original: .6(300) = 180 ml
in new : .7(400) = 280 ml ----> 100 ml more, of coure we added that
1. I do that kind this way
Consider the whole job to be 1
Dustins rate = 1/2 jobs/hr
sisters rate = 1/3 jobs/h
combined rate = 1/2 + 1/3 = 5/6
time to do 1 job at combined rate
= 1/(5/6)
= 6/5 hrs , which is 1 hr, 20 minutes, which you had
Sure! I'd be happy to help you solve the remaining questions.
Let's go through each question step by step and explain how to find the answers:
1. To find how long it will take Dustin and his sister to shovel the walk together, we need to use the concept of their combined work rates. Dustin takes 2 hours to complete the task, which means his work rate is 1/2 of the job per hour (1 job/2 hours = 1/2 job/hour). Similarly, his sister's work rate is 1/3 of the job per hour.
To compute their combined work rate, we add up their individual work rates: 1/2 + 1/3 = 3/6 + 2/6 = 5/6 job per hour.
Now, to find the time it would take them to complete the job together, we need to invert their combined work rate (since the work rate is jobs per hour) and multiply it by the total job. The total job is considered to be 1 (shoveling the walk).
So, (5/6 job/hour) * (1 hour/job) = 5/6 hour. Converting that to minutes, it becomes (5/6 * 60) = 50 minutes.
Therefore, it will take Dustin and his sister 50 minutes, or 1 hour and 12 minutes together to shovel the walk.
2. To find the amount of water the student needs to add, we need to calculate the difference between the amount of water in the desired solution and the current mixture.
The student wants a solution that is 70% water. If the mixture is 300 ml and consists of 60% water, then the amount of water in the mixture is (60/100) * 300 ml = 180 ml.
The desired solution is 70% water, so the amount of water needed is (70/100) * (300 ml + X), where X is the unknown amount of water to be added.
Setting up the equation: (70/100) * (300 ml + X) = 180 ml
To solve for X, we can multiply both sides by 100/70 to cancel out the fraction:
(300 ml + X) = (180 ml) * (100/70)
300 ml + X = 257.14 ml
Solving for X, we subtract 300 ml from both sides:
X = 257.14 ml - 300 ml = -42.86 ml
So, the student needs to add approximately 42.86 ml of water to achieve the desired solution.
3. Let's solve the equation ((d + 2)/(d + 8)) = ((-6)/(d + 8)).
First, we notice that both sides of the equation have a common denominator of (d + 8). We can cross-multiply to solve for d:
(d + 2) * 1 = (-6) * (d + 8)
d + 2 = -6d - 48
To isolate d, we can add 6d to both sides:
d + 2 + 6d = -6d - 48 + 6d
7d + 2 = -48
Next, subtract 2 from both sides of the equation:
7d + 2 - 2 = -48 - 2
7d = -50
Finally, divide by 7 to solve for d:
d = -50/7
So, the solution is d = -50/7.
4. Let's solve the equation 4/n^2 = 7/n + 2.
First, we can bring all terms onto one side of the equation:
4/n^2 - (7/n + 2) = 0
To combine the fractions, we need a common denominator, which is n^2:
(4 - 7n - 2n^2)/n^2 = 0
Now, we can factor the numerator:
(-2n^2 - 7n + 4)/n^2 = 0
(-2n + 1)(n + 4)/n^2 = 0
To find the possible values for n, we set each factor equal to zero:
-2n + 1 = 0 and n + 4 = 0
Solving each equation separately:
-2n + 1 = 0
-2n = -1
n = -1 / -2
n = 1/2
n + 4 = 0
n = -4
Therefore, the solutions to the equation are n = -4 and n = 1/2.
5. Let's solve the equation 2/t + 4/3t = 4/t + 2.
We can work on getting rid of the fractions by finding a common denominator:
The common denominator here is 3t, so let's rewrite the equation with that denominator:
(2 * 3t / 3t) + (4 * t / 3t) = (4 * 3t / 3t) + (2 * 3t / 3t)
Simplifying the equation:
6t + 4t = 12 + 6t
Combining like terms:
10t = 12 + 6t
Subtracting 6t from both sides:
10t - 6t = 12
4t = 12
Dividing by 4:
t = 12 / 4
t = 3
So, the value of t that satisfies the equation is t = 3.
6. Let's solve the equation ((x - 6)/(x^2-6)) = ((-4)/(x-4)).
Since we have fractions on both sides of the equation, we can cross-multiply to solve for x:
(x - 6) * (x - 4) = (-4) * (x^2 - 6)
Expanding and simplifying the equation:
x^2 - 10x + 24 = -4x^2 + 24
Rearranging all terms on one side and combining like terms:
x^2 + 4x^2 - 10x + 4x - 24 - 24 = 0
5x^2 - 6x - 48 = 0
We can factor the quadratic equation if possible. However, in this case, it cannot be factored easily.
Using the quadratic formula, we can find the solutions:
x = (-(-6) ± √((-6)^2 - 4(5)(-48)))/(2(5))
Simplifying:
x = (6 ± √(36 + 960))/(10)
x = (6 ± √996)/(10)
x = (6 ± 2√249)/(10)
Simplifying further:
x = (3 ± √249)/(5)
So, the solutions to the equation are x = (3 + √249)/5 and x = (3 - √249)/5.
I hope these explanations help! Let me know if you have any further questions or need additional assistance.