On the board

A division problem with two positive numbers was written on the board. David noticed that if he increased the dividend by 2 and the divisor by 7, the quotient would not change. By how much should the divisor be increased so that when the dividend is increased by 3, the quotient still remains the same?

Final Answer:

x =  10.5

Step-by-step explanation:

$$\begin{gathered} a/b=c \\ (a+2)/(b+7)=c \\ a=bc \\ a+2=c(b+7) \\ bc+2=bc+7c \\ 2=7c \\ c=7/2=\frac{7}{2}=3\frac{1}{2}=3.5 \\ (a+x)/(b+3)=c \\ a+x=c(b+3) \\ a+x=c(b+3) \\ a+x=cb+3c \\ a+x=a+3c \\ x=3\cdot c=3\cdot 3.5=\frac{21}{2}=10\frac{1}{2}=10.5 \\ \text{ Verifying Solution: } \\ a=35 \\ b=a/c=35/3.5=10 \\ {t}_1=a/b=35/10=\frac{7}{2}=3\frac{1}{2}=3.5 \\ {t}_2=(a+2)/(b+7)=(35+2)/(10+7)=\frac{7}{2}=3\frac{1}{2}=3.5 \\ {t}_2=(a+x)/(b+3)=(35+10.5)/(10+3)=\frac{7}{2}=3\frac{1}{2}=3.5 \\ {t}_1=t_2=t_3 \end{gathered}$$

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