Basic Formulas: The Fish Tale | Mathematics for Class 5: NCERT PDF Download

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Measurements

• If we consider metre, litre, and gram as units of measurement, the higher units are obtained by adding the prefixes deca (meaning ten), hecto (meaning hundred) and kilo (meaning thousand).
• The lower units are obtained by adding the prefixes deci (meaning tenth), centi (meaning hundredth), milli (meaning thousandth)
  

Measures of length

• Length is the longest extent of anything as measured from end to end.
  
• In the above example length is
• used to measure how long the line is.
  • 10 millimetres (mm) = 1 centimetre
  • 10 centimetres (cm) = 1 decimetre
  • 10 decimetres (dm) = 1 metre
  • 10 metres (m) = 1 decametre
  • 10 decametres (dam) = 1 hectometre
  • 10 hectometres (hm) = 1 kilometre (km)

Measures of mass

• Mass is the measure of amount of matter in an object.
  
• 10 milligrams (mg) = 1 centigram
• 10 centigrams (cg) = 1 decigram
• 10 decigrams (dg) = 1 gram
• 10 grams (g) = 1 decagram
• 10 decagrams (dag) = 1 hectogram
• 10 hectograms (hg) = 1 kilogram (kg)

Measures of capacity
Capacity is the amount a container can hold

• 10 millilitres (ml) = 1 centilitre
• 10 centilitres (cl) = 1 decilitre
• 10 decilitres (dl) = 1 litre
• 10 litres (l) = 1 decalitre
• 10 decalitres (dal) = 1 hectolitre
• 10 hectolitres (hl) = 1 kilolitre (kl)
  

Measures of volume
Volume is the measure of the space taken up by something.

• 1000 cubic millimetres (mm³) = 1 cubic centimetre
• 1000 cubic centimetres (cm³) = 1 cubic decimetre
• 1000 cubic decimetres (dm³) = 1 cubic metre
• 1000 cubic metres (m³) = 1 cubic decametre
• 1000 cubic decametres (dam³) = 1 cubic hectometre
• 1000 cubic hectometres (hm³) = 1 cubic kilometre (km³)
  

Volume = Length (m) X Breadth (m) X Height (m)
Unit of volume is cubic length, i.e., cubic metre (m3), cubic centimetre (cm³), etc 

Example 1: Express 1357.912 metres in higher units and lower units.

1357.912 metre
= 135.7912 decametres
= 13.57912 hectometres
= 1.357912 kilometres
= 13579.12 decimetres
= 135791.2 centimetres
= 1357912 millimetres

Note: Capacity and Volume are different. For example. the capacity of a cylinder is 33 litres while its volume is πR²H (R = radius, H = Height of the cylinder). That is. the amount of gas inside the cylinder is 33 litres but the volume occupied by the Cylinder is πR²H.

Example 2: Express 2 g 3 dg 4cg 5mg in terms of milligrams.

2g = 2000mg
3dg = 300mg
4cg = 40mg
2g 3dg 4cd 5mg = (2000 + 300 + 40 + 5) mg =2345mg

Measurements: Addition

• Add the numbers in each position, if a carry is generated then add the carry with the digits in the next position.

Example 1: Add 2l 3dl 4cl 5ml & 6l 6dl 7cl 8ml

2l 3dl 4cl 5ml + 6l 6dl 7cl 8ml 9l 0dl 2cl 3ml [Ans]

Example 2: Express the above result in terms of ml.

2345ml + 6678ml 9023ml

Note: We cannot add or subtract two different quantities. For example, length (metre) cannot be added with mass (g), mass (g) cannot be added with volume (m³), and volume (m³) cannot be added with length (m) 

• Subtract the number in each position, if a borrow is taken then add it to the next position where subtraction will take place.

Example 1: Subtract 3m³ 4dm³ 1cm³ 5mm³ from 4m³ 3dm³ 2cm³ 1mm³ .

Example 2: Express the above in mm3 and write the result.

Measurements: Multiplication and Division

• When a quantity is multiplied by a number, we are actually adding the same quantity a repeated number of times.
  For example,
  When 15kg is multiplied
  by 3, we are actually adding
  the amount 15kg three times.
• When a quantity is divided by a number, we are actually splitting the quantity.
  For example,
  6 litres of oil is
  to be divided into
  3 separate containers.
  Here, we are dividing 6 by 3.
  which is equal to 2. Thus, each
  container would contain 2 litres of oil.
• Two different quantities can be multiplied or divided. When we do so, we get a new quantity.
  For example,
  consider two quantities
  mass (kg) and volume (m³).
  When we divide these two, we
  Kg/m³ get Kg/m³ is the unit of another quantity called density.

Example 1: Divide 5kg 3m³ by 5m

= 1 kg 3m²

Temperature

• Temperature is the measure of the hotness or coldness of a substance. A thermometer is an instrument which is used to measure temperature.

• Temperature is measured in three scales: Celsius, Fahrenheit, and Kelvin.
  
• Celsius a measurementof temperature on a standard in which 0° is the temperature at which water freezes, and 100° the temperature at which it boils.
  Celsius scale is also
  known as centigrade scale.

Fahrenheit Scale: A temperature scale on which water freezes at 32 degrees Fahrenheit (32°F)  and boils at 212 degrees Fahrenheit (212°F).

Kelvin Scale: A temperature scale that defines absolute zero as 0 degrees. Water freezes at 273 degrees and boils at 373 degrees at this temperature. All objects emit thermal energy or heat unless they have a temperature of absolute zero.

Example 1: Convert 123 degrees Fahrenheit into Celsius and Kelvin.

Celsius = (F-32) x 5/9 = (123-32) x 5/9 =  50.55 degree Celsius Kelvin = Celsius + 273 = 50.55 + 273 = 323.55 K

Example 2: Convert 45 Celsius into Fahrenheit and kelvin.

Fahrenheit = (Celsius x 9/5) + 32 = (45 x 9/5) + 32 = 113 degrees Fahrenheit Kelvin = Celsius + Fahrenheit = 45+273 = 318 K