Underline the following ratios that are equivalent to the ratio $a : b$. Answer $\dfrac{1}{b} : \dfrac{1}{a} \quad a^{2} : b^{3} \quad 3a : 3b \quad a+1 : b+1$ John and Sam each have an amount of money. The ratio John's amount : Sam's amount = 6 : 2. John gives Sam $13. The new ratio John's amount : Sam's amount = 2 : 5. Find how much money John has now. Answer $ ...

1. Check each given ratio by simplifying or comparing with $a:b$.

$\dfrac{1}{b}:\dfrac{1}{a}$ can be multiplied by $ab$ in both terms:

$\dfrac{1}{b}:\dfrac{1}{a}=a:b$

So it is equivalent to $a:b$.

$a^{2}:b^{3}\neq a:b$

in general, so it is not equivalent.

$3a:3b=a:b$

So it is equivalent.

$a+1:b+1\neq a:b$

in general, so it is not equivalent.

1. Represent the original amounts using the given ratio.

Original ratio John : Sam $=6:2=3:1$.

Let John's original amount be $6x$ and Sam's original amount be $2x$.

2. Form expressions for the new amounts after John gives Sam \$13.

After John gives Sam $13,

John has $6x-13$,

Sam has $2x+13$.

3. Use the new ratio to form and solve an equation.

New ratio John : Sam $=2:5$.

$\frac{6x-13}{2x+13}=\frac{2}{5}$

$5(6x-13)=2(2x+13)$

$30x-65=4x+26$

$26x=91$

$x=3.5$

4. Find John's current amount.

John's amount now

$=6x-13$

$=6(3.5)-13$

$=21-13$

$=8$