Estimation

Estimation of Dimensions and Distances

Estimation is making a guess of the nearly correct calculations and distances, weights, prices or capacity of things without the actual measurements or calculations. Estimations help us to have rough idea of the answer when we add, subtract, multiply or divide any given quantity.

Sometimes rounding off and approximations are used in making an estimation.

Examples of estimations are as follows:

1. The mass of my friend is 200km
2. The distance of this school from my house is 3 kilometers
3. The weight of our teacher’s table is 500 grammes

When physical quantities are measured, they need to be expressed in standard units of measurement. A unit is therefore a standard used in the measurement of a physical quantity. These physical quantities are length, mass, time, capacity and even currency.

Distances

This is a linear measure of length. The metric system uses multiples of ten to build up its table measurement. The basic unit of length is metre.

10 millimeter (mm) = 1 centimetre

10 centimeters (cm) = 1 decimetre

10 decimeters (dm) = 1 metre

10 meters (m) = 1 decametre

10 decameters (dam) = 1 hectometre

10 hectometers (hm) = 1 kilometre

The frequently used of them is centimeter (cm), metre (m) and kilometer (km)

Examples:

1. Express the following in metre

(i) 12km (ii) 6hm (iii) 324cm

2. Express the following in kilometre

(i) 4580m (ii) 144hm

Solution

1. To change a given number from another unit to metre, we first know how many of those quantity makes one metre

(i) 12km to metre

First, 1 km = 1000m

Then, 12km = 12 × 1000 = 12000m

Therefore, 12km = 12000m

(ii) 6hm to metre

But 1hm = 100m

Then, 2.6 hm = 2.6 × 100 = 26

Therefore, 2.6hm = 26m

(iii) 324cm to metre

But 100cm = 1m

Then, 324cm = 324 ÷ 100 = 3.24

Therefore, 324cm = 3.24 m

Now, we convert to kilometre

2 (i) 4850m to kilometre

But 1000m = 1 kilometre

Then, 4850m = 4850 ÷ 1000 = 4.850

Therefore, 4850m = 4.850km

(ii) 144hm to kilometre

10hm = 1 kilometre

Then, 144hm = 144 ÷ 10 = 14.4

Therefore, 144hm = 14.4km

CLASS ACTIVITY:

1. Express each of the following in centimetre

(i) 2 m (ii) 5km (iii) 9km and 8m

2. Express each of the following in metre

(i) 3800cm (ii) 3000mm (iii) 2km

3. Express each of the following in kilometre

(i) 12580m (ii) 1250dam (iii) 8420m

The Capacity and Mass of Objects

Capacity of Objects

Capacity is the space that is available to hold something. It is closely related to volume because they are both used to measure space. Volume is actually the space occupied by an object. A cubic measure is the measure of volume and this is illustrated below:

1000 cubic millimetre (mm³) = 1 cubic centimetre (cm³)

1000 cubic centimetre (cm³) = 1 cubic decimetre (dm³)

1000 cubic decimetre (dm³) = 1cubic metre (m³)

Example:

Express the following m³: (i) 3500000cm³  (ii) 1450dm³  (iii) 28000mm³

Solutions:

(i) 3500000cm³ to m³

But 1000cm³ = 1 dm³ and 1000dm³ = 1m³

Therefore, (1000 × 1000) cm³ = 1000000cm³ = 1m³

3500000cm³ = 3500000 ÷ 1000000 = 3.500000m³

Hence, 3500000cm³ = 3.5m³

(ii) 1450dm³ to m³

But 1000dm³ = 1m³

Then, 1450dm³ = 1450 ÷ 1000 = 1.450m³

Hence, 1450dm³ = 1.45m³

(iii) 28000mm³ to m³

But 1000mm³ = 1cm³, 1000cm³ =1dm³ and 1000dm³ = 1m³

Therefore, (1000 × 1000 × 1000) mm³ = 1000000000mm³ = 1m³

28000mm³ = 28000 ÷ 1000000000 = 0.000028000m³

Hence, 28000mm³ = 0.000028000m³

Class Activity: Express the following in m³

(i) 756000cm³  (ii) 65400dm³  (iii) 38500dm³  (iv) 17500cm³  (v) 2580dm³

The unit of capacity is Litre (L), which is equal to one cubic decimetre (i.e 1000cm³ = 1 litre). Its table is given below:

10 millilitres (ml) = 1 centilitre (cl)

1000 millitres = 1litre (l)

100 centilitres = 1 litre

1000 litres = 1m³

Example:

Express each of the following in litres (i) 75ml  (ii) 356cl  (iii) 5.3m³

Solution:

(i) 75ml to litre

But 1000ml = 1 litre

Therefore, 75ml = 75 ÷ 1000 = 0.075litre

Hence, 75ml = 0.075litre

(ii) 356cl to litre

100cl = 1 litre

Therefore, 356cl = 356 ÷ 100 = 3.56litres

(iii) 5.3m³ to litre

But 1m³ = 1000 litres

Therefore, 5.3m³ = 5.3 × 1000 = 5300litres

Hence, 5.3m³ = 5300 litres.

The Mass of Objects

Mass is the quantity of matter in a body. In the metric system, the kilogram (kg) is the base unit of mass. The table is given below:

1 decagramme (dg) = 10 grams (g)

1 hectogramme (hg) = 100 g

1 kilogramme (kg) = 1 000 g

1 tonne (t) = 1 000 kg

100 centigramme (cg) =  1 g

200 milligrams (mg) = 1 carat

1000 milligramme (mg) = 1 g

Example:

Express each of the following in kilograms

(i) 60 carats  (ii) 58 mg  (iii) 2.5 tonnes

Solution:

(i) 60 carats to kilogram

1 carat = 200mg = 200 × 1mg

Now, 60 carats = 60 × 200 = 12000mg

But 1000mg = 1gm and 1000g = 1kg

(1000 × 1000) mg = 1000000mg = 1kg

Therefore, 60 carats = 12000 ÷ 1000000 = 0.012kg

Hence, 60 carats = 0.012kg

(ii) 58mg to kilogram

1000mg = 1g and 1g = 1000kg

Therefore, (1000 × 1000) mg = 1000000mg = 1kg

Now, 58mg = 58 ÷ 1000000 = 0.000058kg

Hence, 58mg = 0.000058kg

(iii) 5 tonnes to kilogram

1 tonne = 1000kg

Therefore, 2.5 tonnes = 2.5 × 1000 = 2500kg

Hence, 2.5 tonnes = 2500kg

Class Activity:

1. Express each of the following in litres

(i) 1800ml  (ii) 96cl  (iii) 880ml  (iv) 4.5m³

2. Express each of the following in kilograms (kg)

(i) 138mg  (ii) 2480g  (iii) 1.5g  (iv) 0.28tonnes

Estimation of Other Things

Time

The base unit of time is second (s). Unlike length and mass, time does not follow the metric system of counting in base ten; but rather, the sexagessimal system (base 60) of the ancient Babylonian number system. The illustration is given below:

60 seconds = 1minute

60 minutes = 1 hour

24 hours = 1 day

7 days = 1week

52 weeks = 1 year

365¼ days = 1 year

366 days = 1 leap year

Examples:

1. Change each of the following to seconds

(i) 5 minutes 45 seconds  (ii) 2 hours 4 minutes 15 seconds

Solutions:

(i) 5 minutes 45 seconds to seconds

But, 1 minute = 60 seconds

Therefore, 5 minutes 45 seconds = (5 × 60secs) + 45 secs

= 300 + 45 = 345 secs

(ii) 2 hours 4 minutes 15 seconds

But 1 hour = 60 mins and 2 hours = 2 × 60 = 120 mins

Also, 1 minute = 60 secs,

Then 120 mins = 120 × 60 = 7200 secs

4 mins = 4 × 60 = 240 secs

Therefore, 2 hr 4mins 15 secs = 7200 + 240 + 15

= 745 seconds

2. Find the number of minutes in the following:

(i) 2½ hours  (ii) 500 secs

Solutions:

(i) 2½ hours to minutes

1 hour = 60 mins

Therefore, 2½ × 60 mins

⁵/₂ × 60 = 5 × 30

= 150 minutes

(ii) 1200 secs to mins

But 60 secs = 1 min

Therefore, 1200secs = 1200 ÷ 60

= 20 mins.

Currency

Currency is the system or type of money used by a particular country.

Estimation is often used in market places and shops. People make good estimate of the cost of the article they want to buy while bargaining. The units of the Nigerian currency are Naira (₦) and kobo (k).

Example

Estimate and calculate the sum of the following amounts:

₦15.50, ₦1.75, ₦135.20, ₦18.10, and ₦12.20

Solution

Estimated Value

First step: Approximate the amounts to the nearest whole number;

₦16, ₦2, ₦135, ₦18, ₦12

Second step: Add them up;

₦16 + ₦2 + ₦135 + ₦18 + ₦12

= ₦183

Accurate Calculation

₦15.50k + ₦1.75k + ₦135.20k + ₦18.10k + ₦12.20k

= ₦182.75k

Estimation of Area

The basic unit of area is the square metre (sq. m or m²). However, smaller spaces are measured in centimetre (cm²) while larger spaces like states or countries are measured in acres, hectares and square kilometers (km²). The basic conversions are given below:

1 hectare = 10 000m² = 100 acres

1m² = 1m × 1m = (100 × 100)cm²

1m² = 10 000cm²

1km² = 1km × 1km

= 1000m × 1000m

= 1 000 000m²

2 000cm² = 20001000m²

=  0.2 m²

Example

Estimate and calculate the number of tiles of size 15cm by 28cm that is needed for a square room which has an area of 4.9m².

Solution

Estimated value:

Actual size of tile = 15cm by 28cm

≅ 20cm by 30cm

Area of the tiles = 20 × 30

= 600cm²

Area of square = 4.9m²

≅ 5m²

But 1m = 100cm

1m² = (100 × 100) cm²

= 10 000 cm²

5m² = 5 × 10 000 cm²

= 50 000 cm²

Number of tiles needed

= area of square roomarea of tiles

= 5000600

= 83.33 tiles

Accurate Calculation:

Area of tiles = 15 × 28

= 420cm²

Area of square room = 4.9m² to cm²

= 4.9 × 10 000

= 49 000cm²

Number of tiles needed

= area of square roomarea of tiles

= 49000420

= 116.67

≅ 117 tiles

CLASS ACTIVITY

1. Convert: (a) 6km²to m² (b) 10 000m² to km²   (c) 0.00025m² to cm².
2. Estimate and calculate the sum of the following amounts: ₦134.50, ₦1.54, ₦34.10, ₦57.90 and ₦138.20
3. An interior decorator bought a piece of cloth 2.6m by 3m. Estimate how many pieces he will buy to cover a wall of 195m².

PRACTICE QUESTIONS

1. The actual age of a hospital patient is 39 years, 8 months, 29 days. What is the patient’s age in:  (i) months only?  (ii) days only?
2. A rectangular garden 920m long and 450m wide is to be shared equally among 5 boys. Find:  (i) How many hectares each boy gets  (ii) share of each boy in area
3. If 8 litres of palm oil costs #7.20k, find the for half a litre of this oil.
4. A car travels at 60km per hour. How far does it travel in 30 seconds?
5. A car travels 264km in 2h 30mins. Express the speed in;  (a) km per mins  (b) km per hour

ASSIGNMENT

1. How many 45cl bottles of water can fill a container of 13.5 litres?
2. A man buys 4 litres of petrol for ₦60.00, how much will he pay to fill his car tank which is 52 litres.
3. An interior decorator bought a piece of cloth 2.6m by 3m. Estimate how many pieces he will buy to cover a wall of 195m².
4. Biola’s mother bought 225 eggs for #15.75k. 7 eggs were broken. Find the new average cost per egg for the remaining eggs. Give your answer in kobo
5. Express:  (a) 1728g in kg  (b) 3.5tonnes in kg  (c) 4.5m³ in litres  (d) 14 hectares in m²  (e) 3km 5hm and 4m in metres