Find a and b

Question

Find a and b 

3:a:5=8:4:b

a:b:4=6:3:2

Answer 1

Hi Dylan,

 

In solving proportions containing multiple ratios, we need to solve for each unknown in a step wise manner using the appropriate corresponding ratio pairs to get our solution for each unknown.  We'll solve each of our two multiple ratio problems step-by-step and then check each problem's solution by verifying that each corresponding ratio pair yields the same quotient.

 

 

Problem #1 (Solution):

 

    1.  3:a:5 = 8:4:b (given proportion)

 

    2.  3/a = 8/4 (solve for "a" using first ratio pair):

         (3)(4) = (a)(8)

         8a = 12

           a = 12/8

           a = [4(3)]/[4(2)]

           a = 3/2

 

    3.  a:5 = 4:b (solve for "b" using last ratio pair):

         (3/2):5 = 4:b (plug in "a" value from step 2)

         (3/2)(b) = (5)(4)

         3b/2 = 20

         3b = (20)(2)

         3b = 40

           b = 40/3

 

 

Problem #1 (Check):

 

    3:a:5 = 8:4:b (given proportion)

    3:(3/2):5 = 4:4:(40/3) (solution plug-ins)

 

    3/(3/2) = 8/4 (quotient for first ratio pair)

    3(2/3) = 2

    6/3 = 2

     2 = 2 (same quotient)

 

    (3/2)/5 = 4:(40/3) (quotient for last ratio pair)

    (3/2)(1/5) = 4(3/40)

    [3(1)]/[2(5)] = 12/40

    3/10 = [4(3)]/[4(10)]

    3/10 = 3/10 (same quotient) 

 

 

Problem #2 (Solution):

 

    1.  a:b:4 = 6:3:2 (given proportion)

 

    2.  b/4 = 3/2 (solve for "b" using last ratio pair):
         (b)(2) = (4)(3)
         2b = 12

         b = 12/2

         b = 6

 

    3.  a:b = 6:3 (solve for "a" using first ratio pair):
         a/6 = 6:3 (plug in "b" value from step 2)
         (a)(3) = (6)(6)
         3a = 36
         a = 36/3

         a = 12

 

 

Problem #2 (Check):

    a:b:4 = 6:3:2 (given proportion)
    12:6:4 = 6:3:2 (solution plug-ins)

    12/6 = 6/3 (first ratio pair)
    2 = 2 (same quotient) 

   

    6/4 = 3/2 (last ratio pair)

    [2(3)]/[2(2)] = 3/2

    3/2 = 3/2 (same quotient)

 

 

Since we see that each corresponding ratio pair yields the same quotient in all cases, we are confident that all our answers are correct.

 

Thanks for submitting these problems & glad to help.

 

God bless, Jordan.