Three numbers

The product of three natural numbers is 600. If we reduced one factor by 10, the product would decrease by 400. If we increased one factor by 5 instead, the product would increase to twice the original value. Which three numbers have this property?

Final Answer:

a =  15
b =  5
c =  8

Step-by-step explanation:

$$\begin{gathered} abc=600 \\ (a-10)bc=600-400 \\ a(b+5)c=2\cdot 600 \\ 600=2\times 2\times 2\times 3\times 5\times 5=2^3\times 3\times 5^2 \\ (a-10)(600/a)=600-400 \\ (b+5)(600/b)=2\cdot 600 \\ (a-10)\cdot 600=200a \\ (a-10)\cdot 600=200\cdot a \\ 400a=6000 \\ a=\frac{6000}{400}=15 \\ a=15 \\ (b+5)\cdot 600=2\cdot 600b \\ (b+5)\cdot 600=2\cdot 600\cdot b \\ 600b=3000 \\ b=\frac{3000}{600}=5 \\ b=5 \end{gathered}$$

$$\begin{gathered} c=\frac{600}{a\cdot b}=\frac{600}{15\cdot 5}=8 \\ \text{ Verifying Solution: } \\ {s}_1=a\cdot b\cdot c=15\cdot 5\cdot 8=600 \\ {s}_2=(a-10)\cdot b\cdot c=(15-10)\cdot 5\cdot 8=200 \\ {s}_3=a\cdot (b+5)\cdot c=15\cdot (5+5)\cdot 8=1200 \end{gathered}$$

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