The ratio of the ages of two children is 2: 3. if the younger child is 4 years old, how old is the other child

 The other child is "6 years old". 

Explanation:


Set up the ratio as a fraction: 

  →  " 2/3 = 4/x " ;  Solve for "x"  (the age; in "years", of the "other child") ;  

Note:  GIven the ratios of two children is: 2:3  ; and there is a younger child among the two children;  the number "2" would represent the "younger child; since there are 2 (two) children;  and there are 2 (two) numbers in the given ratios; and among the 2 (two) numbers given in the ratio  — which are:  "2" and "3" ;  the given number: "2" is LESS THAN the given number "3" ; 

As such:

 
  →   " 2/3 = 4/x " ;  Solve for "x" ;  

Cross-multiply; 

2x = 3*4 ; 

2x = 12 ;

Divide each side of the equation by "2" ; 
 to isolate "x" on one side of the equation; & to solve for "x" ; 

2x/2 = 12/2 ; 

to get:

" x = 6 "

Alternate method:


Set up the ratio:


   →  " 2/3 = 4/x " ;  Solve for "x" ;   

Recognize that:  "2/3 = 4/6" ;  and that:  "x = 6" ; 

Alternate method:

Set up the ratio:


   →  " 2/3 = 4/x " ;  Solve for "x" ;   

In the "numerator":  → " 2 * [what value?] = 4 ? "
   → To find that value:  " 4 ÷ 2 = "2 " . 

As such:  Multiply the given:  "3" in the "denominator" by "2" ; to get: 

"3 * 2 = "x" ;   ↔  " x = 6 " . 

 The other child is "6 years old". 

Check our  

The given ratio:  2/3 =? 4/6 ?  Yes! 

{You can even check this using a calculator—if you wish.}.