12. The ratio of the number of pupils in a field to the number of pupils in the hall is $$ 7: 4 $$. There are 168 more pupils in the field than in the hall. How many pupils are in the hall? $$ \begin{array}{ll} ext { (1) } 224 & ext { (2) } 392 \ ext { (3) } 504 & ext { (4) } 672\end{array} $$

Question

12. The ratio of the number of pupils in a field to the number of pupils in the hall is $$ 7: 4 $$. There are 168 more pupils in the field than in the hall. How many pupils are in the hall? $$ \begin{array}{ll}	ext { (1) } 224 & 	ext { (2) } 392 \ 	ext { (3) } 504 & 	ext { (4) } 672\end{array} $$

UpStudy AI · Accepted Answer

Let the number of pupils in the hall be $$ h $$ and the number of pupils in the field be $$ f $$. According to the problem, the ratio of pupils in the field to those in the hall is $$ \frac{f}{h} = \frac{7}{4} $$. This can be expressed as $$ f = \frac{7}{4}h $$.
We also know that there are 168 more pupils in the field than in the hall, which gives us the equation $$ f = h + 168 $$.
Now we can substitute $$ f $$ from the ratio into the second equation:
Substituting gives us: 
$$
\frac{7}{4}h = h + 168
$$
Now, we will solve for $$ h $$.
Solve the equation by following steps:
- step0: Solve for $$h$$:
  $$\frac{7}{4}h=h+168$$
- step1: Move the variable to the left side:
  $$\frac{7}{4}h-h=168$$
- step2: Subtract the terms:
  $$\frac{3}{4}h=168$$
- step3: Multiply by the reciprocal:
  $$\frac{3}{4}h\times \frac{4}{3}=168\times \frac{4}{3}$$
- step4: Multiply:
  $$h=224$$
The number of pupils in the hall is $$ 224 $$.

The ratio of the number of pupils in a field to the number of pupils in the hall is
. There are 168 more pupils in the field than in the hall. How many pupils are
in the hall?

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The ratio of the number of pupils in a field to the number of pupils in the hall is . There are 168 more pupils in the field than in the hall. How many pupils are in the hall?