Working with fractions class 7 ganita prakash

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Working with fractions class 7

Fraction Fusion: The Ultimate Guide to Multiplying & Dividing

Welcome to your complete guide for Chapter 8! This chapter unlocks the methods for working with fractions, moving beyond addition and subtraction to master multiplication and division. Let’s learn how to fuse these numbers together and take them apart with confidence.

1. Multiplication of Fractions

🔑 Key Concept: How to Multiply Fractions

Multiplying fractions is straightforward. You don’t need a common denominator. The rule is:

(Numerator × Numerator) / (Denominator × Denominator)

For a fraction and a whole number, you can write the whole number as a fraction with a denominator of 1. For example, 5 is the same as 5/1.

Questions from Pages 173-176

(Page 173) How far can the tortoise walk in 3 hours? (1/4 km per hour)

Distance = 3 × (1/4) = 3/1 × 1/4 = (3×1)/(1×4) = 3/4 km.

(Page 174) How far can Aaron walk in 2/5 hours? (3 km per hour)

Distance = (2/5) × 3 = (2/5) × (3/1) = (2×3)/(5×1) = 6/5 km (or 1 1/5 km).

(Page 176) Figure it Out – Word Problems:
  1. Tenzin’s Milk: He drinks 1/2 glass per day.
    • In a week (7 days): 7 × (1/2) = 7/2 glasses or 3 1/2 glasses.
    • In January (31 days): 31 × (1/2) = 31/2 glasses or 15 1/2 glasses.
  2. Water Canal: 1 km in 8 days -> 1/8 km per day.
    • In 5 days: 5 × (1/8) = 5/8 km.
  3. Manju’s Oil: 5 litres shared by 3 families.
    • Each family gets: 5/3 litres per week.
    • In 4 weeks: 4 × (5/3) = 20/3 litres or 6 2/3 litres.
  4. Safia’s Moon: Sets 5/6 hour later each day. From Monday 10 pm to Thursday 10 pm is 3 full days.
    • Total delay = 3 days × (5/6) hour/day = 15/6 hours = 5/2 hours or 2.5 hours.
    • The moon will set 2.5 hours after 10 pm on Thursday, which is 12:30 am on Friday.
  5. Multiply and Convert to Mixed Fraction:
    • (a) 7 × 3/5 = 21/5 = 4 1/5
    • (b) 4 × 1/3 = 4/3 = 1 1/3
    • (c) 9/7 × 6 = 54/7 = 7 5/7
    • (d) 13/11 × 6 = 78/11 = 7 1/11

2. Simplifying Before You Multiply

💡 Pro-Tip: Cancel Common Factors First!

Before you multiply, look for any common factors between any numerator and any denominator. You can “cancel” them out by dividing both by their common factor. This makes the numbers smaller and the final multiplication much easier.

Example: Simplifying `12/7 × 5/24` (Page 182)

Step 1: Identify Common Factors. Look at the numerator `12` and the denominator `24`. They have a common factor of 12.
Step 2: Cancel. Divide both by 12.
  • `12 ÷ 12 = 1`
  • `24 ÷ 12 = 2`
Step 3: Multiply the Simplified Fractions.

The problem becomes: (1/7) × (5/2) = 5/14. This is much simpler than calculating (12×5)/(7×24) = 60/168 and then simplifying.

Figure it Out (Page 183)

1. Water Tank: Fills 7/10 of the tank per hour.
  • (a) In 1/3 hour: (1/3) × (7/10) = 7/30 of the tank.
  • (b) In 2/3 hour: (2/3) × (7/10) = (1/3) × (7/5) [canceling 2 and 10] = 7/15 of the tank.
  • (e) For the tank to be full: This is a division problem. Time = 1 (full tank) ÷ (7/10) = 1 × (10/7) = 10/7 hours.
2. Somu’s Land: Government took 1/6. Remaining land = 1 – 1/6 = 5/6.
  • (a) Krishna gets 1/3 of the remaining: (1/3) × (5/6) = 5/18 of the original land.
  • (b) Bora gets 1/3 of the remaining: (1/3) × (5/6) = 5/18 of the original land.
  • (c) Somu keeps: 5/6 (remaining) – 5/18 (Krishna) – 5/18 (Bora) = 15/18 – 10/18 = 5/18 of the original land.
3. Area of rectangle (3 3/4 ft by 9 3/5 ft):

Convert to improper fractions: 15/4 ft and 48/5 ft.
Area = (15/4) × (48/5).
Cancel common factors: 15 and 5 (by 5), 48 and 4 (by 4).
Area = (3/1) × (12/1) = 36 square feet.

3. Division of Fractions

🔑 Key Concept: “Keep, Change, Flip”

You never actually divide fractions. Instead, you turn the division problem into a multiplication problem using the reciprocal.

  • Reciprocal: The “flipped” version of a fraction. The reciprocal of `a/b` is `b/a`.

The rule for division is:

  1. Keep the first fraction the same.
  2. Change the division sign (÷) to a multiplication sign (×).
  3. Flip the second fraction to its reciprocal.

Then, multiply as usual.

Example: Evaluating `2/3 ÷ 5/3` (Page 188)

Step 1: Keep the first fraction: `2/3`.
Step 2: Change the sign: `×`.
Step 3: Flip the second fraction (`5/3`) to its reciprocal: `3/5`.
Step 4: Multiply. The problem is now `2/3 × 3/5`. We can cancel the 3s.
Result: 2/5.

Figure It Out (Page 196)

1. Evaluate the divisions:
  • `3 ÷ 7/9` = 3/1 × 9/7 = 27/7
  • `14 ÷ 2` = 7
  • `2/3 ÷ 2/3` = 2/3 × 3/2 = 1
  • `1/5 ÷ 1/9` = 1/5 × 9/1 = 9/5
2. Choose the correct expression:
  • (a) Maria’s Lace: Total lace is 8m. Each bag uses 1/4m. How many bags? This is division: Total ÷ Amount per item.
    Expression: `8 ÷ 1/4`. This is option (iii).
    Value: 8 × 4/1 = 32 bags.
  • (b) Ribbon for Badges: Total ribbon is 1/2m. It’s used for 8 badges. How much for each? Total ÷ Number of items.
    Expression: `1/2 ÷ 8`. This is option (iv).
    Value: 1/2 × 1/8 = 1/16 meter per badge.
  • (c) Baker’s Flour: Total flour is 5kg. Each loaf needs 1/6kg. How many loaves? Total ÷ Amount per item.
    Expression: `5 ÷ 1/6`. This is option (iii).
    Value: 5 × 6/1 = 30 loaves.

4. Final Word Problems & Puzzles

Questions from Pages 197-198

Q6. Car’s distance: 16 km per litre. How far on 2 3/4 litres?

Distance = (2 3/4) × 16 = (11/4) × 16.
Cancel 4 and 16.
Distance = 11 × 4 = 44 km.

Q8. Mariam’s Cake: Finished 4/5 of the cake. Remaining = 1 – 4/5 = 1/5 of the cake.

This remaining 1/5 is shared equally by 3 friends.
Each friend gets: (1/5) ÷ 3 = (1/5) × (1/3) = 1/15 of the original cake.

Q10. Fraction of shaded square:

The square is divided into 4 equal quadrants. The action is in the bottom-left quadrant, which is 1/4 of the whole area.
That quadrant is divided in half by a diagonal. The shaded part is in one half. Area = 1/2 of 1/4 = 1/8.
That half-quadrant (a triangle) is divided into 4 smaller equal triangles by lines to the midpoint. The shaded part is one of these 4 small triangles.
So, the shaded area is 1/4 of the triangle’s area.
Shaded Area = (1/4) of (1/8) of the whole square = 1/32 of the whole square.

Q12. Pattern `(1 – 1/2) × (1 – 1/3) × …`

Let’s simplify each term:
`(1 – 1/2) = 1/2`
`(1 – 1/3) = 2/3`
`(1 – 1/4) = 3/4`

The product is `(1/2) × (2/3) × (3/4) × … × (1 – 1/n)`.
Notice the cancellation pattern: the numerator of each term cancels with the denominator of the previous term.
(1/2) × (2/3) × (3/4) ...
The only terms left will be the numerator of the first fraction (1) and the denominator of the last fraction.
So for a product up to `(1 – 1/n)`, the result is `1/n`.
For the question up to `(1-1/10)`, the result is 1/10.