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Mathematics: Class 5 — Chapter 4: We the Travellers—II

4.1 Making Sums Equal

In this section, we explore how to balance two groups of numbers so that their sums are equal by interchanging the minimum number of pairs between them.

Example of Balancing Sums

Consider two groups of numbers:

  • Group A: $1, 2, 7, 9$ (Sum = $19$)
  • Group B: $3, 4, 5, 9$ (Sum = $21$)

By interchanging the number $2$ from Group A with the number $3$ from Group B, the new groups become:

  • New Group A: $1, 3, 7, 9$ (Sum = $20$)
  • New Group B: $2, 4, 5, 9$ (Sum = $20$)

The sums are now equal with just one move.

4.2 Fuel Arithmetic & Place Value Addition

Different vehicles require different quantities of fuel. We can use place value charts to add these quantities and understand regrouping.

Addition with Regrouping

When adding numbers, we align them by their place values: Hundreds (H), Tens (T), and Ones (O).

If the sum in the Ones column is $10$ or more, we regroup $10\text{ Ones} = 1\text{ Ten}$.

Example: Add $28\text{ litres}$ and $75\text{ litres}$ of fuel.

$$\begin{array}{c|c|c} \mathbf{H} & \mathbf{T} & \mathbf{O} \\ \hline 1 & 1 & \\ & 2 & 8 \\ + & 7 & 5 \\ \hline 1 & 0 & 3 \end{array}$$

The total quantity of fuel is $103\text{ litres}$.

4.3 Relationship Between Addition and Subtraction

Addition and subtraction are inverse operations. For any addition statement, we can write corresponding subtraction statements.

Examples

  • If $46 + 21 = 67$, then $67 - 21 = 46$ and $67 - 46 = 21$.
  • If $198 - 98 = 100$, then $100 + 98 = 198$ and $198 - 100 = 98$.
4.4 Sums of Consecutive Numbers

Numbers that follow one another in order without skipping are called consecutive numbers (e.g., $1, 2, 3, 4, 5$).

Patterns in Sums

  • Sum of 2 consecutive numbers: Always an odd number. The difference between successive sums is $2$.
    Example: $1 + 2 = 3$, $2 + 3 = 5$, $3 + 4 = 7$.
  • Sum of 3 consecutive numbers: The sum is always $3 \times \text{middle number}$. The difference between successive sums is $3$.
    Example: $1 + 2 + 3 = 6$, $2 + 3 + 4 = 9$, $3 + 4 + 5 = 12$.
  • Sum of 4 consecutive numbers: The difference between successive sums is $4$.
    Example: $1 + 2 + 3 + 4 = 10$, $2 + 3 + 4 + 5 = 14$.
4.5 Adding and Subtracting Large Numbers

When working with large numbers (up to 5 digits), we align digits from right to left (Ones under Ones, Tens under Tens, etc.) and perform addition or subtraction with regrouping.

Addition Example

Find the total length of the North-South Corridor if one section is $1,855\text{ km}$ and the other is $1,862\text{ km}$:

$$\begin{array}{c|c|c|c} \mathbf{Th} & \mathbf{H} & \mathbf{T} & \mathbf{O} \\ \hline 1 & 1 & & \\ 1 & 8 & 5 & 5 \\ +\quad 1 & 8 & 6 & 2 \\ \hline 3 & 7 & 1 & 7 \end{array}$$

Total length = $3,717\text{ km}$.

Subtraction Example

Find the remaining distance of a $2,700\text{ km}$ sea route after travelling $1,083\text{ km}$:

$$\begin{array}{c|c|c|c} \mathbf{Th} & \mathbf{H} & \mathbf{T} & \mathbf{O} \\ \hline & 6 & 9 & 10 \\ 2 & 7 & 0 & 0 \\ -\quad 1 & 0 & 8 & 3 \\ \hline 1 & 6 & 1 & 7 \end{array}$$

Remaining distance = $1,617\text{ km}$.

4.6 Quick Sums and Differences

Mental math strategies can help find sums and differences quickly without standard column calculations.

Piku's Method for Making 100 or 1,000

To find what to add to $32$ to make $100$:

  1. Add a number to make the sum of digits in each column equal to $9$ (i.e., $32 + 67 = 99$).
  2. Add $+1$ to the final result to get $100$ (i.e., $67 + 1 = 68$).
  3. Therefore, $32 + 68 = 100$.

Quickly Subtracting 9 or 99

To subtract $9$ from a number, subtract $10$ and add $1$. To subtract $99$, subtract $100$ and add $1$.

  • $67 - 9 = (67 - 10) + 1 = 57 + 1 = 58$
  • $187 - 99 = (187 - 100) + 1 = 87 + 1 = 88$
4.7 Palindrome Numbers & Grid Puzzles

Palindrome Numbers

A palindrome number is a number that reads the same from left to right as it does from right to left.

  • Examples: $22$, $363$, $404$, $8,558$.
  • Palindromes between $100$ and $200$: $101, 111, 121, 131, 141, 151, 161, 171, 181, 191$.

Grid Puzzles

Arranging numbers $1$ to $9$ in a $3 \times 3$ grid such that each row and column follows increasing (inc) or decreasing (dec) order constraints.

4.8 Even and Odd Numbers

Numbers can be classified based on whether they can be fully paired.

Properties of Even and Odd Sums

  • $\text{Even} + \text{Even} = \text{Even}$ (e.g., $12 + 6 = 18$)
  • $\text{Odd} + \text{Odd} = \text{Even}$ (e.g., $13 + 9 = 22$)
  • $\text{Odd} + \text{Even} = \text{Odd}$ (e.g., $7 + 12 = 19$)