Math Fundamentals: Exploring Ratios

$\dfrac{6}{10} = \dfrac{15}{{\blue{\bigstar}} + 2} \quad \implies \quad {\blue{\bigstar}} + 2 = \dfrac{15 \times 10}{6} \quad \implies \quad {\blue{\bigstar}} + 2 = 25 \quad \implies \quad \boxed{{\blue{\bigstar}} = 23}$

$\frac{6}{10}$ $=$ $\frac{3}{15}$ $=$ $\frac{15}{25}$

$25 = \bigstar + 2$

$\fbox{23} = \bigstar$

$\begin{aligned} \frac 6{10} & = \frac {15}{\blue \bigstar + 2} \\ \frac 35 & = \frac {15}{\blue \bigstar + 2} & \small \blue{\text{Multiply both sides by }\frac 53} \\ 1 & = \frac {25}{\blue \bigstar + 2} \\ \blue \bigstar + 2 & = 25 \\ \implies \blue \bigstar & = 25 - 2 \\ & = \boxed {23} \end{aligned}$

In order for the two fractions to be equal, one must be scaled up or down to equal the other, meaning that both the numerator and denominator must be multiplied by the same value: $6 \times 2.5 = 15\quad \text{ and } \quad 10 \times 2.5 = 25.$ Therefore, the star plus $2$ equals $25,$ so the star has a value of $23.$