1. ## Ratio question

Points A, B, C and D are placed in that order on a line so that AB = 2BC = CD. Express
BD as a fraction of AD.

If the radius of a circle is increased by two units, find the ratio of the new circumference to the new diameter.

There are more but I think if I get the hang of these I will get the rest. Could someone help me to do these

2. Originally Posted by jgv115
Points A, B, C and D are placed in that order on a line so that AB = 2BC = CD. Express
BD as a fraction of AD.

If the radius of a circle is increased by two units, find the ratio of the new circumference to the new diameter.

There are more but I think if I get the hang of these I will get the rest. Could someone help me to do these
If you label the distances between the points on the line...

AB=2x
BC=x
CD=2x

$\displaystyle\frac{BD}{AD}=\frac{3x}{5x}\Rightarro w\ BD=\frac{3}{5}AD$

3. Originally Posted by jgv115

If the radius of a circle is increased by two units, find the ratio of the new circumference to the new diameter.
The ratio of the circumference of a circle to it's diameter is $\pi$

$\displaystyle\frac{{\pi}D}{D}=\frac{2{\pi}R}{2R}=\ pi$

$\displaystyle\frac{2{\pi}(R+2)}{2(R+2)}=\pi$

4. Hello, jgv115!

$\text{Points }A, B, C\text{ and }D\text{ are placed in that order on a line}$

$\text{so that: }\: AB = 2BC = CD.\;\text{ Express }BD\text{ as a fraction of }AD.$

If you had made a simple sketch, you could "eyeball" Archie's solution.

. . $\begin{array}{cccccccccc}
\bullet & ---- & \bullet & -- & \bullet & ---- & \bullet \\
A & 2x & B & x & C & 2x & D \end{array}$

$\text{If the radius of a circle is increased by two units,}$
$\text{find the ratio of the new circumference to the new diameter.}$

This is a trick question.

The ratio of the circumference of any circle to its diameter is $\pi.$

5. Thanks so much guys!

There is one more I found difficulty with:

If a : b = 3 : 4 and a : (b + c) = 2 : 5, find the ratio a : c.

How do I do this?

6. Originally Posted by jgv115
Thanks so much guys!

There is one more I found difficulty with:

If a : b = 3 : 4 and a : (b + c) = 2 : 5, find the ratio a : c.

How do I do this?
A quick way is....

$\displaystyle\frac{a}{b}=\frac{3}{4}\Rightarrow\fr ac{a}{b}=\frac{6}{8}$

$\displaystyle\frac{a}{b+c}=\frac{2}{5}\Rightarrow\ frac{a}{b+c}=\frac{6}{15}$

Therefore the numbers are in the ratio $a:b:c=6:8:7$

Hence

$a:c=6:7$

7. Hello, jgv115!

Another approach . . .

$\text{If }a\!:\! b \,=\, 3\!:\!4\,\text{ and }\,a\!:\!(b + c) \,=\, 2\!:\!5,\,\text{ find the ratio }a\!:\!c.$

We are given: . $\begin{Bmatrix} \dfrac{b}{a} &=& \dfrac{4}{3} & [1] \\ \\[-3mm] \dfrac{b+c}{a} &=& \dfrac{5}{2} & \Rightarrow & \dfrac{b}{a} + \dfrac{c}{a} &=& \dfrac{5}{2} & [2] \end{Bmatrix}$

Substitute [1] into [2]: . $\displaystyle \frac{4}{3} + \frac{c}{a} \:=\:\frac{5}{2} \quad\Rightarrow\quad \frac{c}{a} \:=\:\frac{7}{6}$

Therefore: . $a\!:\!c \:=\:6\!:\!7$

8. Ahh!! Ok, it seems so easy now

Thanks for all the help guys!!

9. Alright, I've gotten into questions that has to do with ratio of volumes and areas.

How would I go about doing this one:

Three similar jugs have heights 8 cm, 12 cm and 16 cm. If the smallest jug holds $1/2$ litre, find the capacities of the other two.

10. Originally Posted by jgv115
Alright, I've gotten into questions that has to do with ratio of volumes and areas.

How would I go about doing this one:

Three similar jugs have heights 8 cm, 12 cm and 16 cm. If the smallest jug holds $1/2$ litre, find the capacities of the other two.
If the jugs were "elongated" versions of each other, their volumes would be in the ratio 8:12:16 or 2:3:4

However, they are "similar", so they also expand in width and depth as the height changes
such that they are magnified versions of each other.

$12=8+4=1.5(8)$

$16=2(8)$

$8cm(Wcm)(Dcm)k=0.5\;\;litres$

$12cm(1.5Wcm)(1.5Dcm)k=1.5(8cm)1.5(Wcm)1.5(Dcm)k=(0 .5\;\;litres)[1.5]^3$

$16cm[2Wcm][2Dcm]k=2(8cm)2(Wcm)2(Dcm)k=(0.5\;\;litres)[2]^3$

$\displaystyle\left[\frac{3}{2}\right]^3=\frac{27}{8}$

The volumes are in the ratio $1:\frac{27}{8}:8=8:27:64=2^3:3^3:4^3$

The capacities of the others can then be found.