Need for Smaller Units

3.1 The Need for Smaller Units

Sonu's mother was repairing a toy. She was trying to join two pieces with a screw. Sonu was closely observing. His mother could not fit the screw because it was not the right size. She took another screw from the box — but Sonu noticed the two screws looked identical in length. Yet when he held them together very carefully, he saw that they were actually of very slightly different lengths.

Sonu was fascinated by how such a small difference in length could matter so much. He wondered how we can know the difference in lengths. He was curious about how little the difference was, because the screws looked almost the same length.

When we need to compare or measure very small lengths, whole-number units^? like centimetres are not enough. We must use smaller units of measurement.

When the ruler is divided between two whole centimetres into 10 equal parts, each small part is \(\frac{1}{10}\) of one centimetre. Counting how many tenths a length contains gives a far more accurate reading.

The length of screw 1 was \(2\frac{1}{10}\) cm (read as two and one-tenth centimetres), and the length of screw 2 was \(2\frac{7}{10}\) cm (read as two and seven-tenth centimetres).

3.2 A Tenth Part

Look at a pencil whose length lies at the fourth small mark past 3 cm on a ruler divided into tenths. Its length is \(3\frac{4}{10}\) units, which can also be written as \((3 \times 1) + \left(4 \times \frac{1}{10}\right)\) units.

Since 10 tenths make one whole, the expression \(3\frac{4}{10} = \frac{34}{10}\), read as thirty-four one-tenths.

\(34 \times \frac{1}{10} = \underbrace{\frac{1}{10}+\frac{1}{10}+\cdots}_{34\text{ times}} = 3 + 1 \times \frac{1}{10} \times 4 = 3\frac{4}{10}\)

A few fraction-in-tenths examples read aloud:

• \(4\frac{1}{10}\) → "four and one-tenth" or "four-one tenths"
• \(\frac{4}{10}\) → "four-tenths"
• \(\frac{41}{10}\) → "forty-one tenths" or "forty-one one-tenths"
• \(41\frac{1}{10}\) → "forty-one and one-tenth"

Working with Mixed Tenths — Sonu's Arm Length

Sonu is measuring his arm. His lower arm is \(2\frac{7}{10}\) units long, and his upper arm is \(3\frac{5}{10}\) units. What is the total length of his arm?

Combining \(2 + 3 = 5\) whole units, and \(\frac{7}{10}+\frac{5}{10}=\frac{12}{10}=1\frac{2}{10}\). So the arm is \(5+1\frac{2}{10} = 6\frac{2}{10}\) units.

Alternatively, write both as tenths: \(2\frac{7}{10}=\frac{27}{10}\) and \(3\frac{5}{10}=\frac{35}{10}\). Sum \(=\frac{62}{10}=6\frac{2}{10}\) units.

Figure it Out (Section 3.2)