A. The number of people working on a certain job varies inversely as the number of days needed to finsih the job. If it takes 10 carpenters to finish a house in 8 days, in how many days will 4 carpenters finish it?
B. The length of a rectangle with a constant area varies inversely as its width. The length of this rectangle is 8in when its width is 3 inches. Find the length when the width is 4 inches.
1. common sense solution:
if it takes 10 carpenters to do a house in 8 days, then 1 carpenter would take 80 days, and 4 carpenters would take 20 days.
2. by the "constant" method
number of days = k/number of carpenters
given:
8 = k/10
k = 80
so number of days = 80/number of capenters
so for 4 carpenters,
number of days = 80/4 = 20
3. the inverse proportion way:
n1 of carpenters/n2 of carpenters = days2/days1
10/4 = days2/8
days2 = 8(10)/4 = 20
notice the actual arithmetic or calculation is the same for all 3 methods.
Pick one of the above ways to do the 2nd question.
A. Well, this sounds like a classic case of "the more, the merrier" or in our case, "the less, the longer it takes." Let's break it down. If it takes 10 carpenters to finish a house in 8 days, we can say that the number of people working is inversely proportional to the number of days needed. To find out how long it would take 4 carpenters to finish the job, we need to set up a proportion.
We can set it up like this: 10 carpenters/8 days = 4 carpenters/x days.
Now let's cross multiply to find "x" (the number of days it would take 4 carpenters):
10 carpenters * x days = 4 carpenters * 8 days.
Dividing both sides by 4 carpenters gives us:
x days = (10 carpenters * 8 days) / 4 carpenters.
Simplifying, we get: x days = 80 days / 4 carpenters.
So, it would take 4 carpenters 20 days to finish the job.
B. Ah, the length and width of a rectangle playing hide and seek with each other. But don't worry, we'll find them. We know that the length of the rectangle varies inversely as its width, which means that as the width goes up, the length goes down and vice versa.
Given that the length is 8 inches when the width is 3 inches, we can set up the following proportion:
Length 1 / Width 1 = Length 2 / Width 2.
Plugging in the values we have, we get:
8 inches / 3 inches = Length 2 / 4 inches.
Now let's cross multiply:
8 inches * 4 inches = 3 inches * Length 2.
Dividing both sides by 3 inches gives us:
(8 inches * 4 inches) / 3 inches = Length 2.
Simplifying, we find:
(32 inches) / 3 inches = Length 2.
So, when the width is 4 inches, the length of the rectangle would be approximately 10.67 inches. Why approximately? Because calculating humorously precise numbers is just not my thing!
A. To solve this problem, we can use the inverse variation formula:
k = (x * y),
where k is the constant of variation, x is the number of people, and y is the number of days.
Given that it takes 10 carpenters to finish a house in 8 days, we can write the equation as:
k = (10 * 8)
Solving for k, we get:
k = 80
Now, we can determine the number of days it will take for 4 carpenters to finish the same job:
k = (4 * y)
Substituting the value of k, we have:
80 = (4 * y)
Solving for y, we get:
y = 80/4
y = 20
Therefore, it will take 4 carpenters 20 days to finish the job.
B. In this problem, the length of the rectangle is inversely proportional to its width. We can represent this relationship using the inverse variation formula:
k = (x * y),
where k is the constant of variation, x is the length, and y is the width.
Given that the length is 8 inches when the width is 3 inches, we can write the equation as:
k = (8 * 3)
Solving for k, we get:
k = 24
Now, we can find the length when the width is 4 inches:
k = (x * 4)
Substituting the value of k, we have:
24 = (x * 4)
Solving for x, we get:
x = 24/4
x = 6
Therefore, when the width is 4 inches, the length of the rectangle will be 6 inches.
A. To solve this problem, we can use the concept of inverse variation. Inverse variation states that two variables are inversely proportional to each other when their product is constant. In this case, the number of people working on a job is inversely proportional to the number of days needed to finish the job.
Let's denote the number of people as P and the number of days as D. According to the problem, when P = 10, D = 8 (10 carpenters finish the job in 8 days). We can write this as:
P * D = k,
where k is a constant. Substitute the given values to find k:
10 * 8 = k,
k = 80.
Now, we can use the equation to solve for the unknown. We know that when P = 4 (4 carpenters), we need to find the number of days, which we'll call x. So we have:
4 * x = 80,
x = 80 / 4,
x = 20.
Therefore, 4 carpenters will finish the job in 20 days.
B. Similarly, we can use the concept of inverse variation to solve this problem. The length of a rectangle varies inversely with its width, which means their product is constant.
Let's denote the length as L and the width as W. According to the problem, when L = 8, W = 3. We can write this as:
L * W = k,
where k is a constant. Substitute the given values to find k:
8 * 3 = k,
k = 24.
Now, we can use the equation to solve for the unknown. We know that when W = 4 (width is 4 inches), we need to find the length, which we'll call x. So we have:
x * 4 = 24,
x = 24 / 4,
x = 6.
Therefore, when the width is 4 inches, the length of the rectangle is 6 inches.