Mathematics: Class 5 — Chapter 1: We the Travellers—I

1.1 Introduction to Travelling

Human beings have always been interested in travelling. Over the centuries, modes of transport have evolved significantly from walking and animal-drawn carts to modern high-speed vehicles.

Historical Transport: Walking, animal-drawn carts, and early boats/ships.
Modern Transport: Bicycles, motorbikes, cars, buses, trains, ships, aeroplanes, submarines, and spacecraft.

1.2 Reading and Writing Large Numbers

To represent quantities larger than $9,999$, we introduce the ten thousand place ($TTh$) in our place value chart.

1.2.1 Place Value Chart

The relation between places is based on multiplying by $10$ at each step:

$10 \text{ Ones} = 1 \text{ Ten} = 10$
$10 \text{ Tens} = 1 \text{ Hundred} = 100$
$10 \text{ Hundreds} = 1 \text{ Thousand} = 1,000$
$10 \text{ Thousands} = 1 \text{ Ten Thousand} = 10,000$

We use a comma to help us read large numbers easily (e.g., $10,000$).

1.2.2 Indian Place Value System Examples

$1,380 = 1 \text{ Thousand} + 3 \text{ Hundreds} + 8 \text{ Tens} + 0 \text{ Ones}$
$9,123 = 9 \text{ Thousands} + 1 \text{ Hundred} + 2 \text{ Tens} + 3 \text{ Ones}$
$10,458 = 1 \text{ Ten Thousand} + 0 \text{ Thousands} + 4 \text{ Hundreds} + 5 \text{ Tens} + 8 \text{ Ones}$ (Ten thousand four hundred fifty-eight)
$45,867 = 4 \text{ Ten Thousands} + 5 \text{ Thousands} + 8 \text{ Hundreds} + 6 \text{ Tens} + 7 \text{ Ones}$ (Forty-five thousand eight hundred sixty-seven)

1.3 Number Patterns and Sequences

Practicing number sequences helps in understanding place value increments:

Adding Hundreds: $456 \rightarrow 567 \rightarrow 678 \rightarrow 789$
Adding Thousands: $10,100 \rightarrow 10,200 \rightarrow 10,300 \rightarrow \dots \rightarrow 10,900$
Adding Tens: $10,105 \rightarrow 10,125 \rightarrow 10,145 \rightarrow \dots$

1.4 Rounding Off Numbers

Rounding numbers helps in estimating quantities to the nearest tens, hundreds, or thousands.

1.4.1 Nearest Tens (10s)

To find the nearest ten of a number, look at the ones digit. If it is $5$ or more, round up; otherwise, round down.

Example: $2,346$ is closer to $2,350$ than to $2,340$. Thus, the nearest ten is $2,350$.

1.4.2 Nearest Hundreds (100s)

To find the nearest hundred, look at the tens digit.

Example: $2,346$ is closer to $2,300$ than to $2,400$. Thus, the nearest hundred is $2,300$.

1.4.3 Nearest Thousands (1,000s)

To find the nearest thousand, look at the hundreds digit.

Example: $2,346$ is closer to $2,000$ than to $3,000$. Thus, the nearest thousand is $2,000$.

1.5 Travelling, Now and Then

Different modes of transport cover different distances in an hour:

On foot: $3\text{--}5 \text{ km}$
By cycle: $12\text{--}20 \text{ km}$
By motorbike: $40\text{--}60 \text{ km}$
By train: $40\text{--}160 \text{ km}$
By aircraft: $750\text{--}920 \text{ km}$
By spacecraft: Minimum $28,000 \text{ km}$

1.6 Pastime Mathematics (Puzzles)

1.6.1 River Crossing Puzzle

A boatman must transport a lion, a sheep, and a bundle of grass across a river. He can only take one at a time. If left alone, the lion eats the sheep, or the sheep eats the grass. How can he cross safely?

1.6.2 Two-Digit Subtraction Puzzle

Take any two different digits, make the largest and smallest 2-digit numbers, and subtract them. Repeat the process with the digits of the difference. The process always ends in the single digit $9$.

1.6.3 King's Horses Puzzle

A caretaker arranges $20$ horses in a square stable so that there are exactly $5$ horses on each side. When a thief steals horses, the caretaker rearranges the remaining horses (e.g., $18$ horses) so that there are still exactly $5$ horses along each side, tricking the king.