Question

Q5: A map is drawn to a scale of 1 : 25000: 13 (a) Two villages are 4.5km apart. Calculate in cm, their distance apart on the map. (b) A housing estate drawn on the map occupies an area of 40cm? Calculate the actual area of the housing estate, giving your answer in km2.​

Answer 1

Given that:

A map is drawn to a scale of 1 : 25000.

• (a) Two villages are 4.5km apart.
• (b) A housing estate drawn on the map occupies an area of 40cm².

To Find:

• (a) Their distance apart on the map in cm.
• (b) The actual area of the housing estate, giving your answer in km².

(a) Converting the distance in cm.

• 1 km = 100000 cm
• 4.5 km = 450000 cm

Let us assume:

• Their distance apart on the map be x.

According to the question.

↠ 1 : 25000 = x : 450000

↠ 1/25000 = x/450000

Cross multiplication.

↠ 450000 = 25000x

↠ 450000/25000 = x

↠ 18 = x

Hence,

• (a) Their distance apart on the map is 18 cm.

(b) Converting the area in km².

• 1 cm² = 0.0000000001 km²
• 40 cm² = 0.000000004 km²

Let us assume:

• The actual area of the housing estate be x.

According to the question.

↠ 1 : 25000 = 0.000000004 : x

↠ 1/25000 = 0.000000004/x

Cross multiplication.

↠ x = 0.000000004 × 25000

↠ x = 0.0001

Hence,

• The actual area of the housing estate is 0.0001 km².

Answer 2

Information provided with us :

• A map is drawn to a scale of 1 : 25000
• Two villages are 4.5km apart

What we have to calculate :

• Their distance apart on the map in cm
• A housing estate drawn on the map occupies an area of 40cm?

Performing Calculations :

First of all in the question it is given that we have to calculate the distance in centimetres. So we would be first changing the unit which is in kilometres into metres

We know that,

• 1 km contains 100000 cm

Changing 4.5km into cm :

$: \longmapsto$ 4.5km = 4.5 × 100000

$: \longmapsto$4.5km = (45/10) × 100000

$: \longmapsto$ 4.5km = 45 × 10000

$: \longmapsto \: \red{\boxed{\bf{4.5km = 45 0000}}}$

• Therefore, 4.5km contains 450000cm

Now let us assume that their distance on the map as an variable. So here we would be assuming it as a. So the ratio would be (a : 450000)

Noticing the question and solving :

$: \longmapsto \: \sf{1 \: : \: 25000 = a :450000 }$

$: \longmapsto \: \sf{ \dfrac{1}{25000} = \dfrac{a}{450000 } }$

$: \longmapsto \: \sf{1 \times 450000 = a \times 25000 }$

$: \longmapsto \: \sf{450000 = 25000 \: a }$

$: \longmapsto \: \sf{25000 \: a \: = \: 450000 }$

$: \longmapsto \: \sf{a\: = \: \dfrac{450000}{25000} }$

$: \longmapsto \: \sf{a\: = \: \cancel\dfrac{450000}{25000} }$

$: \longmapsto \: \sf{a\: = \: \dfrac{450}{25} }$

$: \longmapsto \: \sf{a\: = \: \cancel\dfrac{450}{25} }$

$: \longmapsto \: \sf{a\: = \: \dfrac{90}{5} }$

$: \longmapsto \: \sf{a\: = \: \cancel\dfrac{90}{5} }$

$: \longmapsto \: \boxed{\red{\bf {a\: = \:18 }}}$

$\underline{\bf{Therefore, \: distance \: apart \: on \: the \: map \: in \: cm \: is \: 18cm }}$

_____________________________________

• In this second part of the question we have to calculate the actual area of the housing estate but it should be in kilometre square or km². So first of all we would be converting the area which was in centimetre square or cm² into km²

We know that,

• 1 km² = 10000000000 cm²

Therefore, 1cm² would contain :

$: \longmapsto \: \sf{1cm {}^{2} \: = \: \dfrac{1}{10000000000} }$

$: \longmapsto \: \red{\boxed{\bf{1cm {}^{2} \: = \: 0.0000000001 \: km {}^{2} }}}$

40cm² would contain :

$: \longmapsto \: \sf{40cm {}^{2} \: = \: \dfrac{40}{10000000000} }$

$: \longmapsto \: \red{\boxed{\bf{40cm {}^{2} \: = \: 0.000000004 \: km {}^{2} }}}$

• Now in this step we would be finding out the actual area of the housing estate by assuming the actual area as an variable. Here we would be assuming it as b.

Solving it now by noticing the question :

$: \longrightarrow \: \sf{1 \: : \: 25000 = 0.000000004 \: : \: b}$

$: \longrightarrow \: \sf{ \dfrac{1}{25000} \: = \: \dfrac{0.000000004}{ b}}$

$: \longrightarrow \: \sf{1 \times b =0.000000004 \times 25000 }$

$: \longrightarrow \: \sf{b =0.000000004 \times 25000 }$

$: \longrightarrow \: \boxed{\pink{\bf{b \: = \:0.0001 }}}$

$\underline{\bf{Henceforth, \: actual \: actual \: area \: of \: the \: housing \: estate \: is \:0.0001km{}^{2} }}$