$9$ The number of girls is $\frac{4}{7}$ of the number of boys in a school. $\frac{3}{7}$ of the boys and $\frac{3}{8}$ of the girls obtained Gold Award in a Swim Safer test. $\newline$ $832$ pupils did not obtain Gold Award. $\newline$ (a) How many pupils took part in the Swim Safer test? $\newline$ (b) How many pupils obtained Gold Award?

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Multiply fractions: word problems

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9

The number of girls is

\frac{4}{7}

of the number of boys in a school.

\frac{3}{7}

of the boys and

\frac{3}{8}

of the girls obtained Gold Award in a Swim Safer test.

\newline

832

pupils did not obtain Gold Award.

\newline

(a) How many pupils took part in the Swim Safer test?

\newline

(b) How many pupils obtained Gold Award?

Equation for Girls: Let $b$ represent the number of boys and $g$ represent the number of girls. We have the equation $g = \frac{4}{7}b$ .
Total Number of Pupils: To find the number of pupils who did not get the Gold Award, we need to find the total number of pupils first. Let's call this total $T$ . We know that $T = b + g$ .
Substitute and Simplify: Substitute $g$ from the first equation into the second equation to get $T = b + (\frac{4}{7})b$ .
Fraction of Pupils with Gold Award: Combine like terms to get $T = \left(\frac{7}{7}\right)b + \left(\frac{4}{7}\right)b$ which simplifies to $T = \left(\frac{11}{7}\right)b$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(\frac{3}{7})b + (\frac{3}{14})b$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(\frac{3}{7})b + (\frac{3}{14})b$ . To combine these fractions, find a common denominator, which is $14$ . So, we have $(\frac{3}{8})g$ $0$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(3/7)b$ and for girls, it's $(3/8)g$ . Substitute $g = (4/7)b$ into the equation for girls who got the Gold Award to get $(3/8)(4/7)b$ . Simplify $(3/8)(4/7)b$ to get $(3\times4)/(8\times7)b$ which is $(12/56)b$ . This simplifies further to $(3/14)b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(3/7)b + (3/14)b$ . To combine these fractions, find a common denominator, which is $14$ . So, we have $(3/8)g$ $0$ . Add the fractions to get $(3/8)g$ $1$ . This represents the number of pupils who got the Gold Award.
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(\frac{3}{7})b + (\frac{3}{14})b$ . To combine these fractions, find a common denominator, which is $14$ . So, we have $(\frac{3}{8})g$ $0$ . Add the fractions to get $(\frac{3}{8})g$ $1$ . This represents the number of pupils who got the Gold Award. We know that $(\frac{3}{8})g$ $2$ pupils did not get the Gold Award, which means $(\frac{3}{8})g$ $3$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(\frac{3}{7})b + (\frac{3}{14})b$ . To combine these fractions, find a common denominator, which is $14$ . So, we have $(\frac{3}{8})g$ $0$ . Add the fractions to get $(\frac{3}{8})g$ $1$ . This represents the number of pupils who got the Gold Award. We know that $(\frac{3}{8})g$ $2$ pupils did not get the Gold Award, which means $(\frac{3}{8})g$ $3$ . Substitute $(\frac{3}{8})g$ $4$ from the earlier equation to get $(\frac{3}{8})g$ $5$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(\frac{3}{7})b + (\frac{3}{14})b$ . To combine these fractions, find a common denominator, which is $14$ . So, we have $(\frac{3}{8})g$ $0$ . Add the fractions to get $(\frac{3}{8})g$ $1$ . This represents the number of pupils who got the Gold Award. We know that $(\frac{3}{8})g$ $2$ pupils did not get the Gold Award, which means $(\frac{3}{8})g$ $3$ . Substitute $(\frac{3}{8})g$ $4$ from the earlier equation to get $(\frac{3}{8})g$ $5$ . Find a common denominator for the fractions on the left side of the equation, which is $14$ . So, we have $(\frac{3}{8})g$ $7$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(\frac{3}{7})b + (\frac{3}{14})b$ . To combine these fractions, find a common denominator, which is $14$ . So, we have $(\frac{3}{8})g$ $0$ . Add the fractions to get $(\frac{3}{8})g$ $1$ . This represents the number of pupils who got the Gold Award. We know that $(\frac{3}{8})g$ $2$ pupils did not get the Gold Award, which means $(\frac{3}{8})g$ $3$ . Substitute $(\frac{3}{8})g$ $4$ from the earlier equation to get $(\frac{3}{8})g$ $5$ . Find a common denominator for the fractions on the left side of the equation, which is $14$ . So, we have $(\frac{3}{8})g$ $7$ . Subtract the fractions to get $(\frac{3}{8})g$ $8$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(\frac{3}{7})b + (\frac{3}{14})b$ . To combine these fractions, find a common denominator, which is $14$ . So, we have $(\frac{3}{8})g$ $0$ . Add the fractions to get $(\frac{3}{8})g$ $1$ . This represents the number of pupils who got the Gold Award. We know that $(\frac{3}{8})g$ $2$ pupils did not get the Gold Award, which means $(\frac{3}{8})g$ $3$ . Substitute $(\frac{3}{8})g$ $4$ from the earlier equation to get $(\frac{3}{8})g$ $5$ . Find a common denominator for the fractions on the left side of the equation, which is $14$ . So, we have $(\frac{3}{8})g$ $7$ . Subtract the fractions to get $(\frac{3}{8})g$ $8$ . Multiply both sides by $(\frac{3}{8})g$ $9$ to solve for $g = (\frac{4}{7})b$ $0$ : $g = (\frac{4}{7})b$ $1$ .
Calculate Number of Boys: Now, let's find the number of boys and girls who obtained the Gold Award. For boys, it's $(\frac{3}{7})b$ and for girls, it's $(\frac{3}{8})g$ . Substitute $g = (\frac{4}{7})b$ into the equation for girls who got the Gold Award to get $(\frac{3}{8})(\frac{4}{7})b$ . Simplify $(\frac{3}{8})(\frac{4}{7})b$ to get $(\frac{3\times4}{8\times7})b$ which is $(\frac{12}{56})b$ . This simplifies further to $(\frac{3}{14})b$ . Add the fractions of boys and girls who got the Gold Award to find the total fraction of pupils who got the Gold Award: $(\frac{3}{7})b + (\frac{3}{14})b$ . To combine these fractions, find a common denominator, which is $14$ . So, we have $(\frac{3}{8})g$ $0$ . Add the fractions to get $(\frac{3}{8})g$ $1$ . This represents the number of pupils who got the Gold Award. We know that $(\frac{3}{8})g$ $2$ pupils did not get the Gold Award, which means $(\frac{3}{8})g$ $3$ . Substitute $(\frac{3}{8})g$ $4$ from the earlier equation to get $(\frac{3}{8})g$ $5$ . Find a common denominator for the fractions on the left side of the equation, which is $14$ . So, we have $(\frac{3}{8})g$ $7$ . Subtract the fractions to get $(\frac{3}{8})g$ $8$ . Multiply both sides by $(\frac{3}{8})g$ $9$ to solve for $g = (\frac{4}{7})b$ $0$ : $g = (\frac{4}{7})b$ $1$ . Calculate $g = (\frac{4}{7})b$ $1$ to get $g = (\frac{4}{7})b$ $3$ .