Question

in a concert hall, there was ⅗ as many men as women. there were 56 more children than women. the number of women was 25% of the total number of people in the concert hall. how many people were there in the concert hall?​

Answer 1

Let's use algebra to solve this problem step by step. Let:

M = the number of men

W = the number of women

C = the number of children

We are given several pieces of information:

The number of men is ⅗ as many as women, so M = (⅗)W.

There were 56 more children than women, so C = W + 56.

The number of women was 25% of the total number of people, so W = 0.25(M + W + C).

Now, we can set up a system of equations based on these conditions:

Equation 1: M = (⅗)W

Equation 2: C = W + 56

Equation 3: W = 0.25(M + W + C)

We need to solve this system of equations to find the values of M, W, and C.

First, let's express Equation 1 in terms of W:

M = (⅗)W

Now, substitute this expression for M into Equation 3:

W = 0.25(((⅗)W) + W + C)

Next, simplify Equation 3:

W = 0.25((⅗)W + W + C)

Multiply both sides by 4 to eliminate the fraction:

4W = ((⅗)W + W + C)

Now, get rid of the fractions by multiplying both sides by 5:

20W = (3W + 5W + 5C)

Combine like terms:

20W = 8W + 5C

Now, we have two equations:

Equation 2: C = W + 56

Equation 4: 20W = 8W + 5C

Let's solve Equation 2 for C:

C = W + 56

Now, substitute this expression for C into Equation 4:

20W = 8W + 5(W + 56)

Distribute the 5 on the right side:

20W = 8W + 5W + 280

Combine like terms:

20W = 13W + 280

Subtract 13W from both sides:

7W = 280

Now, divide by 7:

W = 280 / 7

W = 40

So, there were 40 women in the concert hall. Now, we can find the number of men using Equation 1:

M = (⅗)W

M = (⅗) * 40

M = 24

Now, we can find the number of children using Equation 2:

C = W + 56

C = 40 + 56

C = 96

Finally, to find the total number of people in the concert hall, add the number of men, women, and children:

Total = M + W + C

Total = 24 + 40 + 96

Total = 160

There were 160 people in the concert hall.