Question

The points A, B, C and D lie in order on a straight line. AB : BD = 3:4 AC: CD = 3:2 Work out AB : BC: CD Give your answer in its simplest form.​

Answer 1

The ratio $|AB|:|BC|:|CD|$ in its simplest form is $15:6:14$

Let

$|AB|=x\\|BC|=y\\|CD|=z$

from the question, $|AB|:|BD|=x:(y+z)=3:4\text{ or}$

$\frac{x}{y+z}=\frac{3}{4}\\\\4x=3y+3z$

also, from the question, $|AC|:|CD|=(x+y):z=3:2\text{ or}$

$\frac{x+y}{z}=\frac{3}{2}\\\\2x+2y=3z$

Now we have the equation set

$4x=3y+3z\\2x+2y=3z$

eliminating $3z$, and rearranging, we get

$4x=3y+(2x+2y)\\2x=5y\\\frac{x}{y}=\frac{5}{2}$

substituting $5y$ for $2x$ in the equation $2x+2y=3z$, then rearranging, gives

$(5y)+2y=3z\\7y=3z\\\frac{z}{y}=\frac{7}{3}$

We have obtained the ratios $x:y=5:2$, and $z:y=7:3$, we now have to form equivalent fractions for each ratio, so that we can obtain $x:y:z$

$\frac{x}{y}\text{ and }\frac{z}{y}=\frac{5}{2}\text{ and }\frac{7}{3}\\\\=\frac{5\times 3}{2\times 3}\text{ and }\frac{7\times 2}{3\times 2}\\\\=\frac{15}{6}\text{ and }\frac{14}{6}$

What we have now is

$x:y=5:2=15:6\\z:y=7:3=14:6$

The final ratio in its simplest form is

$x:y:z=15:6:14\\\text{ or}\\|AB|:|BC|:|CD|=15:6:14$

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