Question:Find the ratio of `17 1/2` new pence to £`1.12 1/2` as a fraction in its lowest terms. 

`(17 1/2)/(112 1/2)=35/225=7/45.`

Question:Find the ratio of 3.15 m to 27 cm. 

`(3.15 m)/(27 cm) = (315 cm)/(27 cm) = 35/3`
The ratio is 35:3

Question:Express `17 1/2` new pence as a percentage of £2.50. 

`(17 1/2)/250 = 35/500 = 7/100 = 7/100 * 100% = 7%`

Question:Find 15% of 3 meters. 

`15/100 xx 3 m = 45/100 m = 45/100 xx 100 cm = 45 cm.`

Question:Find 8% of £24.18 correct to the nearest new penny. 

`8/100 xx £24.18`

= `(£193.44)/100`

= £1.9344

= £1.93 (correct to nearest new penny).

Question:Find 73% of £24.18 

First method
`73/100 xx £24.18` 

= `(£1765.14)/100`

= £17.6514

= £17.65 (correct to nearest new penny).Second method
73% of £24.18 = £17.6514 = £17.65 (to nearest new penny)

Question:A father leaves a legacy of £5500 to be divided between his three daughters, Ann, Beryl and Caroline in the ratios 2:4:5. Find how much each receives. 

First Method
If legacy is divided so that Ann receives 2 equal parts, Beryl 4 equal parts and Caroline 5 equal parts, the total must be divided into (2+4+5) equal parts, or 11 equal parts:
`1/11` of £5500=£500.
 Ann receives `2xx£500=£1000` .
Beryl receives `4xx£500=£2000` .
Caroline Receives `5xx£500=£2500` .Second method
Suppose that Ann receives £2k, Beryl £4k and Caroline £5k.
Then 2k+4k+5k=5500.
=>11k=5500 or k=500.
Ann receives £1000, Beryl £2000 and Caroline £2500.

Question:The lengths of the sides of the triangles ABC and XYZ are shown in figure. Are the triangles similar? If so, find an angle equal to the angle A.

Figure Triangle: 

XY:XZ:YZ=6:8:10=3:4:5=AB:BC:AC
 Therefore the triangles ABC and YXY are similar (corresponding sides are XY and AB; XZ and BC; YZ and AC).
 The angle A is opposite BC. It therefore equals the angle opposite XZ, i.e. the angle Y.

Question:In the triangles ABC and XYZ, the angle A equals the angle Y and the angle B equal the angle X. Given that AB=6 cm, AC=4 cm, BC=7 cm and that XY=4 cm, find the lengths of XZ and YZ. 

Since `hat(A)=hat(B)` and  `hat(B) = hat(X)` the angles C and Z are equal.
Therefore the triangles `{("ABC"),("YXZ"):}` are similar.
`:. (AB)/(YX)=(BC)/(XZ)=(AC)/(YZ)`
`:. 6/4=7/(XZ)=4/(YZ)`
`:. XZ=4 2/3 cm; YZ=2 2/3 cm`

Question:In the following fig. XY is parallel to BC. Prove that the triangles ABC and AXY are similar, and hence find two expressions equal to `(XY)/(BC)`. 

figure Line 

Since XY is parallel to BC,
              `hat(X)=hat(B)` (corresponding).
Similarly  `hat(Y)=hat(C)`.
The triangle `{("AXY"),("ABC"):}` are equiangular and therefore similar.

`:. (XY)/(BC)=(AX)/(AB)=(AY)/(AC)`

(Notice that any ratio of lengths on AB is equal to the corresponding ratio of lengths on AC. For example, `(AX)/(XB)=(AY)/(YC)`. The ratio of the transversals, XY and BC is, however, equal only to the ratio of a side of the small triangle AXY to the corresponding side of the large triangle ABC.)