A Peek Beyond the Point · Lesson 6 of 11
Units of Measurement
“Use decimal place value to express the same length, mass, or money amount in different units.”
• Convert lengths between millimetres, centimetres, metres, and kilometres. • Convert masses between milligrams, grams, and kilograms. • Convert amounts between paise and rupees. • Explain conversion direction using the size of the chosen unit. • Check that a converted measurement represents the same quantity.
Length Conversion
A measurement combines a number and a unit. The same pencil can be 12 cm long or 120 mm long because a millimetre is a smaller measuring part. Changing the unit changes the count, while the pencil itself stays the same length. Unit relationships tell us how to perform that change.
One centimetre contains ten millimetres. Therefore one millimetre is one-tenth of a centimetre, or 0.1 cm. To write a millimetre measurement in centimetres, group millimetres in tens. To write centimetres in millimetres, count ten millimetres for each centimetre.
Problem
Express 12 mm in centimetres and 5.6 cm in millimetres.
- 1.12 mm = 10 mm + 2 mm. The first group is 1 cm and the remainder is 0.2 cm.
- 2.Thus 12 mm = 1.2 cm.
- 3.For 5.6 cm, the 5 whole centimetres contain 50 mm and 0.6 cm contains 6 mm.
- 4.Therefore 5.6 cm = 56 mm. The larger numerical count belongs to the smaller unit.
A metre contains one hundred centimetres. A centimetre is therefore one-hundredth of a metre. Fifteen centimetres contain fifteen of those hundredth-metre parts, so they equal 0.15 m. A metre also contains 1000 millimetres, making a millimetre one-thousandth of a metre.
| Unit relationship | Smaller unit in larger-unit form |
|---|---|
| 1 cm = 10 mm | 1 mm = 0.1 cm |
| 1 m = 100 cm | 1 cm = 0.01 m |
| 1 m = 1000 mm | 1 mm = 0.001 m |
| 1 km = 1000 m | 1 m = 0.001 km |
Problem
Convert 4 cm and 325 cm to metres.
- 1.Each centimetre is 1/100 m, so 4 cm = 4/100 m = 0.04 m. The zero in tenths is necessary.
- 2.325 cm = 300 cm + 25 cm.
- 3.The first part is 3 m and the second is 25/100 m = 0.25 m.
- 4.Thus 325 cm = 3.25 m. Conversely, 2.07 m is 200 cm + 7 cm = 207 cm.
The decimal position follows from the unit relationship. It is not a guess based on how the number looks. A kilometre contains 1000 metres and each metre contains 1000 millimetres. Counting millimetres in a kilometre therefore gives 1000 × 1000 = 1,000,000 millimetres.
Compare a ruler’s smallest interval with 0.1 mm, 0.05 mm, and 1 mm. A hair or thin paper may be smaller than an ordinary ruler can measure accurately. Use these as size comparisons, and do not treat a drawn picture as a real measuring instrument unless its scale is specified.
Weight Conversion
In everyday language, people often speak of an object’s weight in grams or kilograms. These units measure mass. The chapter uses the familiar heading Weight Conversion, and the numerical unit relationships are straightforward: one kilogram contains one thousand grams, and one gram contains one thousand milligrams.
Problem
Convert 5 g, 10 g, and 254 g to kilograms.
- 1.5 g consists of 5 thousandths of a kilogram, so it is 0.005 kg.
- 2.10 g consists of 10 thousandths, equal to 1 hundredth. It is 0.010 kg or 0.01 kg.
- 3.254 g = 200 g + 50 g + 4 g.
- 4.In kilograms these are 0.2 kg + 0.05 kg + 0.004 kg, giving 0.254 kg.
The same relationship works in reverse. A mass of 2.5 kg contains 2 complete kilograms and half a kilogram. These parts contain 2000 g and 500 g, so the mass is 2500 g. Milligrams work similarly: 1 mg = 0.001 g, and 5 mg = 0.005 g.
Problem
Rice packets and heaps have masses 10 kg, 1 kg, 100 g, 10 g, and 1 g. Express their combined mass in kilograms.
- 1.Convert the gram quantities: 100 g = 0.1 kg, 10 g = 0.01 kg, and 1 g = 0.001 kg.
- 2.The kilogram quantities are 10, 1, 0.1, 0.01, and 0.001.
- 3.Add corresponding places to obtain 11.111 kg.
- 4.Notice that each mass in the list is one-tenth of the previous mass. The list illustrates the decimal place-value relationship.
Rupee–Paise Conversion
One rupee contains one hundred paise. A paisa is therefore one-hundredth of a rupee. Decimal money notation records whole rupees to the left of the point and hundredths of a rupee to the right. An amount such as ₹0.05 represents five paise, even though it has a zero immediately after the point.
Problem
Express 75 paise and 250 paise in rupees.
- 1.75 paise = 75/100 rupee = ₹0.75.
- 2.250 paise = 200 paise + 50 paise.
- 3.These parts are ₹2 and ₹0.50, so 250 paise = ₹2.50.
- 4.The two zeros in ₹2.50 do not mean 250 rupees. The point fixes the rupees position.
Ask an adult about prices they remember from childhood, or inspect an old bill, coin, or stamp if one is available. Record any paise amount as a decimal rupee amount. Keep the date with a historical price; a past price is not a claim about what the item costs today.
Moving to a larger unit gives fewer units for the same positive measurement. For example, 68 g = 0.068 kg, not 68,000 kg. Check the unit relationship and whether the resulting size makes sense before accepting a conversion.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
What is 134 mm in centimetres?
What is 0.89 m in centimetres?
What is 68 g in kilograms?
What is ₹0.05 in paise?
What happens to the numerical count when the same positive length is expressed in a smaller unit?
Ten millimetres make a centimetre, so 134 mm contains 13 cm and 4 mm = 13.4 cm.
Practice Problems
- Convert 70 mm and 203.6 cm into centimetres and millimetres respectively.
- Convert 36 cm, 50 cm, and 207 cm into metres.
- Convert 465 g, 704 g, and 1560 g into kilograms.
- Express 0.56 kg in grams and 15 mg in grams.
- Convert 10 paise, 99 paise, and 250 paise into rupees.
- How many millimetres make one kilometre? Explain in two stages.
- A fee is 45 paise per person. What total is collected from 100,000 people at that stated rate?
70 mm = 7 cm because 10 mm make 1 cm. For 203.6 cm, count 10 mm per centimetre: 2030 mm + 6 mm = 2036 mm.
Key Takeaways
• A measurement includes both a number and a unit. • Converting units preserves the physical quantity. • One centimetre contains ten millimetres; one metre contains one hundred centimetres. • One kilogram contains one thousand grams; one gram contains one thousand milligrams. • One rupee contains one hundred paise. • A smaller unit gives a larger numerical count for the same positive quantity. • Use unit relationships and a size check to verify conversions.