Ornamental shrubs

Children committed to planting 240 ornamental shrubs. Their commitment, however, was exceeded by 48 shrubs. Write the ratio (r) of actually planted shrubs and commitment by the lowest possible integers a/b.

Correct answer:

r =  6:5
a =  6
b =  5

Step-by-step explanation:

$$\begin{gathered} x=240 \\ y=x+48=240+48=288 \\ r=y:x=288:240=6:5=\frac{6}{5}=1\frac{1}{5}=1.2 \end{gathered}$$

$$\begin{gathered} 288=2^5\cdot 3^2 \\ 240=2^4\cdot 3\cdot 5 \\ \text{GCD}(288,240)=2^4\cdot 3=48 \\ d=GCD(y,x)=GCD(288,240)=48 \\ a=y/d=288/48=6 \end{gathered}$$

$b=x/d=240/48=5$

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