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The Question states:

Two discs are placed, in contact with each other, on a table. Their radii are 4 cm and 9 cm. An elastic band is stretched round the pair of discs. Calculate

(a) the angle subtended at the centre of the smaller disc by the arc that is in contact with the elastic band.

(b) the length of the part of the band that is in contact with the smaller disc.

(c) the length of the part of the band that is in contact with the larger disc.

(d) the total length of the stretched band.

(Hint. The straight parts of the stretched band are common tangents to the two circles.)

So far I've figured out the angle subtended at the centre of both of the circles by the arc that is in contact with the elastic band is the same, but I could not go further.

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    Geometry problems with a diagram are waaaaay better... – 2017-02-28
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    @DonAntonio No diagram was given :/ so I assumed it just looked like BB-8 with a rubber band around it – 2017-02-28
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    http://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/TangentToTwoCircles.shtml – 2017-02-28

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As DonAntonio said, geometry problems with a diagram are waaaaay better:

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We know that a tangent to a circle is perpendicular to the radius at the point of tangency, so $B_1C_1\parallel B_2C_2$ and $\triangle{AB_1C_1}\sim\triangle{AB_2C_2}$. Thus, we have $${|AC_1|\over|B_1C_1|}={|AC_2|\over|B_2C_2|}.\tag1$$ Setting $x=|AC_1|$ and substituting $r_1$ and $r_2$ for the two known radius lengths, we get the equation $${x\over r_1}={x+r_1+r_2\over r_2}.\tag2$$ Solving (2) gives $x=r_1{r_2+r_1\over r_2-r_1}$ and so $$\cos{\angle{AC_1B_1}}=\cos{\angle{AC_2B_2}}={r_2-r_1\over r_2+r_1}.\tag3$$ I trust that you can take it from here.