Step 1: Define Variables Based on Ratio

• Let the present ages of A and B be $A$A and $B$B.
• Four years later, the ages of A and B will be A+4 and $\mathrm{}B\; +\; 4$B+4, respectively.
• The given ratio 4 years later is 8:9.
• Let $A\; +\; 4\; =\; 8x$A+4=8x and B+4=9x for some value $x$x.

Step 2: Express Present Ages in Terms of $x$x

• $A\; =\; 8x\; –\; 4$A=8x−4
•  B=9x−4

Step 3: Use Average Age Information

• The average of the present ages of A and B is 47 years:

$\frac{A+B}{2}=47\backslash frac\{A\; +\; B\}\{2\}\; =\; 47$ 

$\mathrm{\Rightarrow \; A}+B=94A\; +\; B\; =\; 94$ 

Step 4: Substitute Present Ages into the Average Age Equation

• Substitute A=8x−4 and  B=9x−4 into  A+B=94:
  •   (8x−4)+(9x−4)=94
    ⇒  17x−8=94
    ⇒  17x=102
    ⇒ x=6

Step 5: Calculate Present Ages and Difference

• Substitute x back to find present ages:
  • A=8x−4=8(6)−4=48−4=44
    B=9x−4=9(6)−4=54−4=50
  • The difference in present ages of A and B is:  B−A=50−44=6

The difference in present ages of A and B is 6 years.

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