Find the ratio of male teachers to male students

Question

At a school function, the ratio of teachers to students is $5:18$. The ratio of female students to male students is $7:2$. If the ratio of the female teachers to female students is $1:7$, find the ratio of the male teachers to male students.

Answer 1

Lets consider there are 7 female student and 2 male student.(7:2)

total student = 9

teacher : student = 5:18

therefore for 18 student there will be 14 girls and 4 boy students

5:14:4

now only 1 female teacher is required for 7 female students (1:7)

here we have 14 female students hence 2 female teacher is required

and total teacher is 5

out of it 2 are female hence 3 teacher are male

5:14:4(teacher:female student:male student)

3:2:14:4(male teacher: female teacher :female student:male student)

hence 3: 4

this is what i found if its wrong please reply

Answer 2

Let's say that there are $T_f$ female teachers, $T_m$ male teachers, $S_f$ female students, and $S_m$ male students. You are given that,

$$ (T_f + T_m) / (S_f + S_m) = 5/18\\ S_f / S_m = 7/2\\ T_f / S_f = 1/7 $$

And what you want to find is,

$$ T_m / S_m $$

You can solve this now by substituting in the things you already know,

$$ T_f = 1/7 \times S_f \\ S_m = 2/7 \times S_f \\ T_m = 5/18 \times (S_f + S_m) - T_f $$

So,

$$ \begin{aligned} T_m/S_m &= T_m / S_f \times S_f / S_m \\ &= (5/18 \times (S_f + S_m) - T_f) / S_f \times 7/2 \\ &= (5/18 \times (S_f + 2/7 \times S_f) - 1/7 \times S_f) / S_f \times 7/2 \\ &= (5/18 \times (1 + 2/7) - 1/7) \times 7/2 \\ &= 3/4 \end{aligned} $$

Answer 3

Let A = # of Female Teachers

Let B = # of Male Teachers

Let X = # of Female Students

Let Y = # of Male Students

Then the statement "the ratio of Teachers to students is 5:18" can be represented by the following equation, EQ1:

EQ1: (A+B)/(X+Y) = 5/18 _________(A+B = number of teachers, X+Y = number of students)

 18*(A+B) = 5*(X+Y) _________(Still EQ1)

 18*A + 18*B = 5*X + 5*Y ____(Still EQ1)

The statement "the ratio of famale students to male students can be represented by the following equation, EQ2:

EQ2: X/Y = 7/2 --> X = (7/2)*Y

The statement "the ratio of female teachers to female students can be represented by the following equation, EQ3:

EQ3: A/X = 1/7 --> A = X/7

We want to solve for the ratio of male teachers (B) to male students (Y)

In order to do that, we want first want to substitute X in EQ3 with the value of X defined in EQ2

A = X/7

A = (7/2)*Y/7 ____________(substituted (7/2)*Y for X)

A = Y/2 ________________ (We will call this EQ4)

Now, we want to substitute X in EQ1 with the value of X defined in EQ2, and A in EQ1 with the value of A defined in EQ4, and solve for B/Y.

18*A + 18*B = 5*X + 5*Y _____________________(EQ1 from above)

18*(Y/2) + 18*B = 5*((7/2)*Y) + 5*Y _________(Substituted for A and X)

9*Y + 18*B = 17.5*Y + 5*Y

18*B = 17.5*Y + 5*Y - 9*Y

18*B = 13.5*Y

B/Y = 13.5/18 = 3/4

Therefore, the ratio of male teachers to male students, B:Y = 3:4.