Working with Fractions · Lesson 8 of 8
Chapter Summary and Practice
“Revise the whole chapter, connect its methods, and apply them to mixed calculations and visual challenges.”
• Recall and explain the multiplication and division rules for fractions. • Choose operations from the meaning of a problem and verify the result. • Predict the size of positive products and quotients. • Solve mixed sharing, measurement, area, and nested-fraction problems. • Use the chapter’s fraction methods to investigate branching and product patterns.
What We Have Studied
This chapter connected fraction calculations with equal parts, areas, and quantities in everyday situations. Use the table to recall the main ideas before doing the mixed practice. If a method feels uncertain, return to its teaching lesson and then try the relevant problem again.
| Lesson | Main idea | Method or useful check |
|---|---|---|
| Multiplying Fractions and Whole Numbers | Repeated copies; finding a fraction of a quantity | For a/b of n, divide n by b and take a shares. |
| Multiplying Two Fractions | A fraction of a fraction; area in a unit square | Count selected cells among all equal cells in the original whole. |
| The Multiplication Rule and Simplification | Multiply numerator factors and denominator factors | Cancel common factors before multiplying when convenient. |
| Understanding the Size of a Product | Multiplication can shrink or enlarge a positive quantity | Compare the multiplier with 1; swapping factor order preserves the product. |
| Division of Fractions and Reciprocals | Division asks for a missing factor | Multiply the dividend by the divisor’s reciprocal; check divisor × quotient. |
| Reasoning Through Fraction Division Problems | Sharing, group counts, areas, and combined rates | Identify the unknown quantity and its unit before choosing an operation. |
| Fractional Relations and Nested Fractions | Track the same whole through repeated shares | Multiply nested fractions or subtract known regions; respect stated conversion rates. |
Essential Relationships
These formulas collect the relationships already explained in the lessons. They are most useful when you can connect each symbol with the quantity in a question. For division, the divisor must be nonzero; the comparison statements apply to positive quantities.
| Operation on a positive quantity | Number between 0 and 1 | Number equal to 1 | Number greater than 1 |
|---|---|---|---|
| Multiply by the number | Result decreases | Result stays the same | Result increases |
| Divide by the number | Result increases | Result stays the same | Result decreases |
Representative Revision Examples
The following short examples revisit common choices rather than introduce new methods. In each one, identify the meaning of the operation, calculate, and then check the answer against that meaning.
Problem
Find 9/10 of 2 2/3 litres.
- 1.Convert 2 2/3 to 8/3 litres. The word “of” gives 9/10 × 8/3.
- 2.Cancel 9 with 3 to leave 3 and 1; cancel 8 with 10 by 2 to leave 4 and 5.
- 3.The result is 12/5 = 2 2/5 litres. Since 9/10 is below 1, this is less than the original 2 2/3 litres.
Problem
A bottle has 1 1/2 litres of juice. Each serving is 1/4 litre. How many servings are available?
- 1.The question asks how many quarter-litre groups fit into the total, so use 3/2 ÷ 1/4.
- 2.Multiply by the reciprocal: 3/2 × 4 = 6.
- 3.There are 6 servings. Check: 6 × 1/4 litre = 3/2 litres.
Problem
Four fifths of a cake is eaten. The rest is shared equally by three friends. What fraction of the original cake does each receive?
- 1.The remaining cake is 1 − 4/5 = 1/5 of the whole.
- 2.Share that remainder: 1/5 ÷ 3 = 1/5 × 1/3 = 1/15.
- 3.Each friend receives 1/15 of the original cake. Three such shares total the 1/5 remainder.
Keep the whole unchanged in nested fractions. Cancel factors rather than terms of sums. Take the reciprocal of the divisor only. Convert measurement units before comparing amounts, and use square units for area.
Quiz
What is 2/3 of 3/4?
Which operation counts 1/6-litre servings in 2 litres?
What is the reciprocal of 1 2/5?
Which statement is correct for positive quantities?
A shaded region is half of a triangle that occupies one eighth of a square. What fraction of the square is shaded?
A student wants 3/4 ÷ 2/5. Which multiplication check verifies the result 15/8?
Practice Problems
- Calculate in lowest form: 5/6 × 9/10; 7/12 × 18/21; 2 1/4 × 4/9.
- Calculate and check by multiplication: 3/5 ÷ 2/7; 4 ÷ 8/9; 2 2/3 ÷ 1 1/3.
- Predict whether the product is above both factors, below both, or between them: 7/6 × 8/5; 3/7 × 4/9; 2/5 × 3.
- A tap fills 3/8 of a tank in one hour. How much is filled in 2/3 hour? How long is needed for one tank?
- Mira’s novel has 400 pages. She reads 1/5 of the whole book yesterday and 3/10 of the whole book today. How many pages remain?
- A car travels 16 km per litre of petrol. How far can it travel on 2 3/4 litres at the same rate?
- A train takes 5 1/6 hours for a journey while a plane takes 1/2 hour. How many hours does the plane save?
- A rectangle measures 2 2/5 m by 1 3/4 m. Find its area.
- A family uses 2/7 of its land for a path. It plants trees on 3/5 of the remaining land. What fraction of the original land has trees?
- Two fountains fill a tank individually in 1/4 day and 1/6 day. Find their combined filling rate and completion time.
- Explain why 565/465 × 707/676 exceeds each factor and also exceeds 1, without calculating the exact product.
- A student claims 4/7 ÷ 1/3 = 4/21. Use an answer-size prediction to identify the mistake, then correct it.
Products: 3/4, 1/2, 1. Quotients: 21/10, 9/2, 2. Product positions: above both, below both, between. Tank: 1/4 filled; a full tank takes 8/3 hours. Book: 400 × (1 − 1/5 − 3/10) = 200 pages. Car: 16 × 11/4 = 44 km. Journey: 31/6 − 1/2 = 28/6 = 4 2/3 hours saved. Area: 12/5 × 7/4 = 21/5 = 4 1/5 m². Trees: 3/5 × 5/7 = 3/7 of the original land. Fountains: 4 + 6 = 10 tanks per day, so 1/10 day. Both large factors exceed 1, so their product exceeds each and exceeds 1. The mistaken division multiplied by 1/3 instead of 3; correct result 4/7 × 3 = 12/7, which exceeds 4/7.
Visual and Pattern Challenges
These challenges combine methods from across the chapter. Sketch intermediate fractions or label the amount arriving at each point. Show the reasoning, because the final number alone does not explain why a complicated-looking picture or product becomes manageable.
- For the nested shaded square above, find the orange fraction of the whole and justify every intermediate area.
- If the orange region in that diagram were shared equally into three regions, what fraction of the whole would each occupy?
- Calculate (1 − 1/2), then (1 − 1/2) × (1 − 1/3), then the product with additional factors (1 − 1/4) and (1 − 1/5). What pattern do you notice?
- Calculate the product (1 − 1/2) × (1 − 1/3) × (1 − 1/4) × (1 − 1/5) × (1 − 1/6) × (1 − 1/7) × (1 − 1/8) × (1 − 1/9) × (1 − 1/10) without multiplying large numbers.
- For a whole number n at least 2, state and explain the value of the product of (1 − 1/k) for every whole number k from 2 through n.
The orange fraction is 1/4 × 1/2 × 3/4 = 3/32. Each of three equal orange shares is 1/32. The successive products are 1/2, 1/3, and 1/5. Rewrite each factor as (k − 1)/k: 1/2 × 2/3 × 3/4 × 4/5. Intermediate numerator-denominator factors cancel, leaving 1/5. Through the denominator 10, the result is 1/10. Through n, cancellation leaves 1/n. This uses the multiplication rule and cancellation already studied.
All ants start below A and move upward. At every labelled point, they split equally among its outgoing paths. What fraction of the original ants reaches the mango tree, and what fraction reaches the sugarcane field? Label fractions at A, B, C, and D before adding paths ending at each source.
All ants reach A. Half go directly to the mango tree and half reach B. At B, half of that arriving half goes to the mango tree and half reaches C: C receives 1/4 of the original group. C has four outgoing paths, so each takes 1/4 × 1/4 = 1/16 of the original group. Two go to mango, one directly to sugarcane, and one to D. D receives 1/16 and splits it into two shares of 1/32. Sugarcane receives 1/16 + 1/32 = 3/32. Mango receives 1/2 + 1/4 + 1/16 + 1/16 + 1/32 = 29/32. The fractions sum to 1, so every ant is accounted for.
A queen attacks along its row, column, and diagonals. On a 4 × 4 board, place four queens so that none attacks another. Then try eight queens on an 8 × 8 board. This chapter-end puzzle develops spatial reasoning; it is a separate challenge after the fraction revision.
For a 4 × 4 board, place one queen in each row in columns 2, 4, 1, and 3, in that order. For an 8 × 8 board, one valid row-by-row column sequence is 1, 5, 8, 6, 3, 7, 2, 4. Each sequence uses different columns, and no pair of queens has a column difference equal in magnitude to its row difference, so no pair shares a diagonal.
Key Takeaways
• Fraction multiplication and division are grounded in equal parts and missing-factor relationships. • Multiply numerator factors and denominator factors; cancel common factors carefully. • For division, multiply by the nonzero divisor’s reciprocal and check the original relationship. • Compare a positive multiplier or divisor with 1 to predict how the answer should change. • Keep the reference whole and units clear in sharing, area, rate, and nested-fraction problems. • Use diagrams and intermediate fractions to explain branching, shaded regions, and cancellation patterns.
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Fractional Relations and Nested Fractions
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