Mathematics: Class 5 — Chapter 6: The Dairy Farm

Find the numbers hiding in all the shapes (numbers are between $1$ and $24$):

• a) $\text{Blue Rectangle} \times \text{Pink Circle} = \text{Purple Triangle}$
• b) $\text{Blue Rectangle} \times \text{Blue Rectangle} = \text{White Hexagon}$
• c) $\text{Yellow Trapezoid} \times \text{Orange Diamond} = \text{Purple Triangle}$
• d) $\text{Orange Diamond} \times \text{Green Pentagon} = \text{Orange Diamond}$
• e) $\text{Yellow Trapezoid} \times \text{Pink Circle} = \text{Blue Semi-circle}$
• f) $\text{Blue Semi-circle} \times \text{Blue Rectangle} = \text{Yellow Trapezoid} \times \text{Purple Triangle}$

• Digit Placement: Place the digits $2$, $5$, and $3$ appropriately to get a product close to $100$.
• Which Number Am I? Find the two-digit number using these clues:
  • I am greater than $8$.
  • I am not a multiple of $4$.
  • I am a multiple of $9$.
  • I am an odd number.
  • I am not a multiple of $11$.
  • I am less than $50$.
  • My ones digit is even.
  • My tens digit is odd.
• Make Your Own Numbers: Choose any two numbers and one operation from the grid to make all numbers between $0$ and $20$. Grid numbers: $100, 25, 5, 10, 2, 36, 12, 4, 3$.

• $4 \times 10 = 40$
• $20 \times 10 = 200$
• $10 \times 40 = 400$
• $10 \times 10 = 100$
• $20 \times 50 = 1,000$
• $80 \times 10 = 800$
• $3 \times 100 = 300$
• $8 \times 100 = 800$
• $10 \times 100 = 1,000$
• $2 \times 1,000 = 2,000$
• $5 \times 1,000 = 5,000$
• $10 \times 1,000 = 10,000$
• $20 \times 1,000 = 20,000$

This strategy involves halving one factor and doubling the other factor to make the multiplication simpler. It works exceptionally well when multiplying by numbers like $5$, $25$, or $50$.

• $18 \times 5 = 9 \times 10 = 90$
• $3 \times 18 = 6 \times 9 = 54$
• $22 \times 5 = 11 \times 10 = 110$
• $14 \times 3 = 7 \times 6 = 42$
• $38 \times 5 = 19 \times 10 = 190$
• $16 \times 4 = 8 \times 8 = 64$
• $35 \times 14 = 70 \times 7 = 490$

Adjust one of the numbers to its nearest multiple of $10$, multiply, and then add or subtract the difference.

• Example (Subtraction): $4 \times 19 = 4 \times (20 - 1) = 80 - 4 = 76$
• Example (Addition): $14 \times 21 = 14 \times (20 + 1) = 280 + 14 = 294$
• Practice:
  • $7 \times 52 = 7 \times (50 + 2) = 350 + 14 = 364$
  • $12 \times 28 = 12 \times (30 - 2) = 360 - 24 = 336$
  • $75 \times 31 = 75 \times (30 + 1) = 2250 + 75 = 2325$
  • $99 \times 15 = (100 - 1) \times 15 = 1500 - 15 = 1485$

Break down both numbers into their place values and multiply them in a grid.

For $12 \times 35$:

• $12 = 10 + 2$
• $35 = 30 + 5$

Sum of products: $300 + 50 + 60 + 10 = 420$

Multiply each place value component step-by-step:

For $35 \times 12$:

• $2 \times 5 = 10$
• $2 \times 30 = 60$
• $10 \times 5 = 50$
• $10 \times 30 = 300$
• Total: $10 + 60 + 50 + 300 = 420$

For $35 \times 12$:

• $2 \times 35 = 70$
• $10 \times 35 = 350$
• Total: $70 + 350 = 420$

The standard column multiplication method using regrouping (carrying over digits to the next place value).

For $268 \times 4$:

• $4 \times 8 = 32$ (Write $2$, carry $3$ tens)
• $4 \times 60 = 240$ plus $30$ carried over $= 270$ (Write $7$, carry $2$ hundreds)
• $4 \times 200 = 800$ plus $200$ carried over $= 1,000$ (Write $10$)
• Total: $1,072$

1. An auditorium has $35$ rows with $42$ seats in each row. How many people can sit in this auditorium?
   Solution: $35 \times 42 = 1,470$ people.
2. Priya jogs $4$ km every day. How many kilometers will she jog in $31$ days?
   Solution: $4 \times 31 = 124$ km.
3. A school received $36$ boxes of books with $48$ books in each box. How many total books did the school receive?
   Solution: $36 \times 48 = 1,728$ books.
4. Priya uses $16$ meters of cloth to make $4$ kurtas. How much cloth would she need to make $8$ kurtas?
   Solution: $8$ kurtas is double of $4$ kurtas, so she needs $16 \times 2 = 32$ meters of cloth.
5. Gollappa has $29$ cows. Each cow produces $5$ litres of milk per day. How many litres of milk do the cows produce in total each day?
   Solution: $29 \times 5 = 145$ litres.
6. A dairy cooperative gets milk from $268$ villagers. Each villager has at least $4$ milk-giving cows. What is the minimum number of cows?
   Solution: $268 \times 4 = 1,072$ cows.
7. The cooperative sells $1$ kg cow ghee for ₹$574$. If they produce $125$ kg of ghee in a month, how much do they earn?
   Solution: $574 \times 125 = $ ₹$71,750$.

• $1 \times 1 = 1$
• $11 \times 11 = 121$
• $111 \times 111 = 12,321$
• $1111 \times 1111 = 1,234,321$

• $11 \times 12 = 132$ (Digits of $12$ split with their sum in the middle: $1$, $1+2=3$, $2$)
• $11 \times 34 = 374$
• $11 \times 56 = 616$
• $11 \times 78 = 858$

• $5 \times 5 = 25$
• $15 \times 15 = 225$
• $25 \times 25 = 625$
• $35 \times 35 = 1,225$
• $45 \times 45 = 2,025$
• $55 \times 55 = 3,025$