The number of red, blue, and green marbles in a bag is in the ratio 3 : 4 : 6. If 20 red marbles, 15 blue marbles, and an unknown number of green marbles are added to the bag, the ratio of red, blue, and green marbles becomes 4 : 5 : 7. Determine the number of green marbles added.
5
Let the original numbers be Red = \(3x\), Blue = \(4x\), Green = \(6x\), and let \(g\) green marbles be added.
After addition: Red = \(3x + 20\), Blue = \(4x + 15\), Green = \(6x + g\).
The new ratio of Red : Blue is given as 4 : 5, so:
\(\dfrac{3x + 20}{4x + 15} = \dfrac{4}{5}\)
\(5(3x + 20) = 4(4x + 15)\) \(\Rightarrow\) \(15x + 100 = 16x + 60\) \(\Rightarrow\) \(x = 40\).
So Red = 3×40 = 120, Blue = 4×40 = 160, Green = 6×40 = 240.
After addition: Red = 140, Blue = 175. Using Blue : Green = 5 : 7:
\(\dfrac{175}{240 + g} = \dfrac{5}{7}\) \(\Rightarrow\) \(5(240 + g) = 7 \times 175 = 1225\) \(\Rightarrow\) \(240 + g = 245\) \(\Rightarrow\) \(g = 5\).
Hence, 5 green marbles were added.
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