Two Numbers Ratio Change

Two numbers are in the ratio 5:6. If one number is smaller by seven and the other is larger by seven, there will be new numbers in the ratio 4:7. What is the original, and what are the changed numbers?

Final Answer:

a =  35
b =  42
c =  28
d =  49

Step-by-step explanation:

$$\begin{gathered} a=5/6\cdot b \\ a-7=4/7\cdot (b+7) \\ 6a-5b=0 \\ 7a-4b=77 \\ Pivot:Row1\leftrightarrow Row2 \\ Row2-\frac{6}{7}\cdot Row1\to Row2 \\ -1.57b=-66 \\ b=\frac{-66}{-1.57142857}=42 \\ a=\frac{77+4b}{7}=\frac{77+4\cdot 42}{7}=35 \\ a=35 \\ b=42 \end{gathered}$$

$b=42$

$c=a-7=35-7=28$

$d=b+7=42+7=49$

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