Work out the value of x if (5x + 2) : (2x + 3) = 7 : 3.

Step-by-step solution

1. 1

   Rewrite the ratio as an equation between fractions. A ratio $a:b$ is the fraction $\tfrac{a}{b}$, so

   $\frac{5x+2}{2x+3}=\frac{7}{3}.$

2. 2

   Cross-multiply. Multiply each numerator by the opposite denominator; brackets are essential:

   $3(5x+2)=7(2x+3).$

3. 3

   Expand both sides.

   $15x+6=14x+21.$

4. 4

   Collect terms. Subtract $14x$ and then $6$:

   $x+6=21\ \Longrightarrow\ x=15.$

5. 5

   Check by evaluating the two quantities. With $x=15$:

   $5x+2=77,\qquad 2x+3=33,$

   so the ratio is $77:33$. Dividing both parts by their common factor $11$ gives $7:3$ ✓ — exactly as required.