Question

A, b and c are positive integers.

a : b = 3 : 8
and
b : c = 6 : 11
work out the smallest possible value of a + b + c

Answer 1

`a:b=3:8\\\\(a)/(b)=(3)/(8)\qquad\text{multiply both sides by b}\\\\\underline{\underline{a=(3)/(8)b}}\\\\b:c=6:11\\\\(b)/(c)=(6)/(11)\qquad\text{cross multiply}\\\\6c=11b\qquad\text{divide both sides by 6}\\\\\underline{\underline{c=(11)/(6)b}}\\\\a+b+c\qquad\text{substitute}\\\\(3)/(8)b+b+(11)/(6)b=(3\cdot3)/(8\cdot3)b+b+(11\cdot4)/(6\cdot4)b=(9)/(24)b+(24)/(24)b+(44)/(24)b=(77)/(24)b`

`\text{If a, b, and c are positive integers, then the sum a+ b + c is positive}\\\text{integer}.\\\\(77)/(24)b\ \text{is positive integer if}\ b=24k\ (k\ \text{is positive integer}).\\\\\text{The smallest possible value of a + b + c is for k = 1}\\\\b=24(1)=24\\\\(77)/(24)\cdot24=77\\\\Answer:\ \boxed{77}`