Question

The points O, P, Q and R all lie on the same line segment, in that order, such that the ratio of OP: PQ : QR is equal to 5:6: 1. If OR = 36, find OQ.

Answer 1

The Ratio OP: PQ: QR = 5: 6: 1 means that if the line segment OR is 5 + 6 + 1 = 12 then

`(OP)/(OR)=(5)/(12)`
`(PQ)/(OR)=(6)/(12)`
`(QR)/(OR)=(1)/(12)`

But, the problem is that OR is not 12, rather, it is 36; therefore, we need to multiply the ratios above by 36:

`\begin{gathered} (OP)/(OR)=(5)/(12)\rightarrow OP=(5)/(12)OR \\ \end{gathered}`
`OP=(5)/(12)*36`
`OP=15`
`(PQ)/(OR)=(6)/(12)\rightarrow PQ=(6)/(12)* OR`
`PQ=(6)/(12)*36`
`PQ=18`
`(QR)/(OR)=(1)/(12)\rightarrow QR=(1)/(12)* OR`
`QR=(1)/(12)*36`
`QR=3`

Hence, we have the lengths OP, PQ, and QR.

The line segment OQ is

`OQ=OP+PQ`

therefore,

`OQ=15+18`
`OQ=33.`

which is our answer!