Mathematics: Class 5 — Chapter 9: Coconut Farm

Consider an array of $35$ coconuts arranged in $5$ rows and $7$ columns:

• Multiplication Fact: $5 \times 7 = 35$
• Division Fact 1: $35 \div 7 = 5$ ($35$ split into $7$ groups has $5$ in each group)
• Division Fact 2: $35 \div 5 = 7$ ($35$ split into $5$ groups has $7$ in each group)

The core components of a division statement are defined as:

• Dividend (N): The number to be divided.
• Divisor (D): The number by which the dividend is divided.
• Quotient (Q): The result obtained from division.

This relationship is expressed as:

$$\text{Dividend (N)} = \text{Divisor (D)} \times \text{Quotient (Q)}$$

Observe the following patterns in division:

• $150 \div 3 = 50$ (since $50 \times 3 = 150$)
• $1000 \div 10 = 100$ (since $10 \times 100 = 1000$)
• $2000 \div 20 = 100$
• $440 \div 44 = 10$ (since $44 \times 10 = 440$)

When dividing numbers by $10$ or its multiples, the digits shift place values:

• $40 \div 10 = 4$ (Ones place)
• $400 \div 10 = 40$ (Tens place)
• $4000 \div 10 = 400$ (Hundreds place)

A large dividend can be split into parts that are easily divisible by the divisor.

Example: Solve $1248 \div 6$

• Split $1248$ into $1200 + 48$.
• We know $1200 \div 6 = 200$.
• And $48 \div 6 = 8$.
• So, $1248 \div 6 = 200 + 8 = 208$.

Example (Subtraction): Solve $1992 \div 4$

• Express $1992$ as $2000 - 8$.
• We know $2000 \div 4 = 500$.
• And $8 \div 4 = 2$.
• So, $1992 \div 4 = 500 - 2 = 498$.

To divide a number by $4$, you can halve the number twice. To divide by $8$, you can halve it three times.

Example: Solve $128 \div 4$

• First half: Half of $128$ is $64$.
• Second half: Half of $64$ is $32$.
• Therefore, $128 \div 4 = 32$.

For any division, the relationship between the components is given by:

$$\text{Dividend (N)} = \text{Divisor (D)} \times \text{Quotient (Q)} + \text{Remainder (R)}$$

Where the Remainder (R) must always be less than the Divisor (D).

Example: $726 \div 4$

• $726 = 4 \times 181 + 2$
• Here, Dividend $N = 726$, Divisor $D = 4$, Quotient $Q = 181$, and Remainder $R = 2$.

This method involves taking away easy multiples of the divisor from the dividend step-by-step.

Example: Solve $535 \div 25$

• Take away $20$ groups of $25$: $535 - 500 = 35$.
• Take away $1$ group of $25$: $35 - 25 = 10$.
• Total Quotient = $20 + 1 = 21$.
• Remainder = $10$.

Example: Solve $324 \div 3$

• Hundreds: $3\text{ Hundreds} \div 3 = 1\text{ Hundred}$.
• Tens: $2\text{ Tens} \div 3$ is not possible without regrouping. Write $0$ in the Tens place of the quotient.
• Regrouping: Regroup $2\text{ Tens}$ into $20\text{ Ones}$. Total ones = $20 + 4 = 24\text{ Ones}$.
• Ones: $24\text{ Ones} \div 3 = 8\text{ Ones}$.
• Final Quotient: $108$.

A mathematical statement is true if both sides of the '=' sign have the same value.

• $8 \times 9 = 70 + 2$ is True ($72 = 72$).
• $20 - 6 = 7 \times 3$ is False ($14 \neq 21$).
• $48 \div 3 = 4 \times 4$ is True ($16 = 16$).

• Odd Numbers: When two odd numbers are added, the sum is always even (e.g., $3 + 5 = 8$).
• Halving: Halving an even number can sometimes lead to an odd number (e.g., half of $10$ is $5$) and sometimes to an even number (e.g., half of $12$ is $6$).
• Multiplication by 5: Multiplying an even number by $5$ always leads to a number with '0' in the Ones place (e.g., $4 \times 5 = 20$).