Question

If y varies directly as x, and y is 180 when x is n and y is n when x is 5, what is the value of n?

Answer 1

(You don’t need to read all of that. N is 30) When
y
=
180
,
x
=
n
. If we write the value of y on the top and x on the bottom, one side of the proportion is:
180
n
2. When
y
=
n
,
x
=
5
. So the other side of the proportion would be:
180
n
=
n
5
3. Cross multiply.
180
(
5
)
=
n
(
n
)

900
=
n
2
4. Take the square root of both sides to find
n
.
√
900
=
√
n
2
30
=
n
That's your answer!
Check to make sure the proportion makes sense.
Does
180
30
equal
30
5
?

Answer 2

`\bf \qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad y=kx\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \textit{we also know that } \begin{cases} y=180\\ x=n \end{cases}\implies 180=kn\implies \cfrac{180}{n}=k`

`\bf therefore\qquad \boxed{y=\cfrac{180}{n}x} \\\\\\ \textit{and we know that } \begin{cases} y=n\\ x=5 \end{cases}\implies n=\cfrac{180}{n}(5)\implies n^2=900 \\\\\\ n=√(900)\implies \blacktriangleright n=30 \blacktriangleleft`