Two numbers are in the ratio 5 to 8. When 2 is added to each, the ratio of the resulting numbers is 2 to 3. Find the numbers.
Two numbers are in the ratio 5 to 8. When 2 is added to each, the ratio of the resulting numbers is 2 to 3. Find the numbers.
Two numbers are in the ratio 5 to 8.
When 2 is added to each, the ratio of the resulting numbers is 2 to 3.
Find the numbers.
\(\begin{array}{|lrcll|} \hline (1) & \dfrac{x}{y} &=& \dfrac{5}{8} \quad & | \quad \cdot y \\\\ & \mathbf{x} &\mathbf{=}& \mathbf{\dfrac{5}{8}\cdot y } \\\\ \hline \\ (2) & \dfrac{x+2}{y+2} &=& \dfrac{2}{3} \quad & | \quad \cdot (y+2) \\\\ & x+2 &=& \dfrac{2}{3} \cdot (y+2) \quad & | \quad \cdot \dfrac{3}{2} \\\\ & \dfrac{3}{2}\cdot(x+2) &=& y+2 \quad & | \quad x =\dfrac{5}{8}\cdot y \\\\ & \dfrac{3}{2}\cdot\left(\dfrac{5}{8}\cdot y+2 \right) &=& y+2 \\\\ & \dfrac{3}{2}\cdot\dfrac{5}{8}\cdot y+2\cdot \dfrac{3}{2}&=& y+2 \\\\ & \dfrac{15}{16}\cdot y+3&=& y+2 \\\\ & y+2 &=& \dfrac{15}{16}\cdot y+3 \quad & | \quad -\dfrac{15}{16}\cdot y \\\\ & y-\dfrac{15}{16}\cdot y+2 &=& 3 \quad & | \quad - 2 \\\\ & y-\dfrac{15}{16}\cdot y &=& 3 - 2 \\\\ & \dfrac{1}{16}\cdot y &=& 1 \quad & | \quad \cdot 16 \\\\ & \mathbf{ y } &\mathbf{=}& \mathbf{16} \\\\ \hline \\ & x &=& \dfrac{5}{8}\cdot y \quad & | \quad y=16 \\\\ & x &=& \dfrac{5}{8}\cdot 16 \\\\ & \mathbf{ x } &\mathbf{=}& \mathbf{10} \\ \hline \end{array}\)
The numbers are 10 and 16
Let x be one number and y the second
We have
x / y = 5 / 8 multiply through by y
x = (5/8)y (1)
Also
(x + 2) / ( y + 2) = 2/3 cross-multiply
3(x + 2) = 2(y + 2) simplify
3x + 6 = 2y + 4 sub (1) in for x
3(5/8 * y) + 6 = 2y + 4 simplify
(15/8 ) y + 6 = 2y + 4 subtract 4, (15/8)y from both sides
2 = (1/8) y multiply through by 8
16 = y and x = (5/8((16) = 10
5x / 8x
(5x+2) / (8x+2) = 2/3 cross multiply
15x+6 = 16x+4
2= x
5x= 10 8x = 16