Fractions · Lesson 9 of 9
Chapter Summary and Practice
“Connect sharing, measuring, equivalence, comparison, and fraction arithmetic through complete-chapter revision.”
• Connect equal shares, parts of a whole, and number-line lengths. • Use numerator and denominator meanings to interpret fractions and mixed numbers. • Choose equivalent fractions for simplification, comparison, and arithmetic. • Solve mixed practical problems and explain the units and reasoning. • Use unit-fraction puzzles to connect comparison, addition, and subtraction.
One idea connects the whole chapter
Fractions began with a fair share of a roti, but the same idea now describes lengths, weights, volumes, and points on a number line. Choose one whole, divide it into equal parts, and count the parts you need. The denominator identifies the fractional unit; the numerator counts copies of it. Nearly every method in this chapter grows from keeping those two jobs clear.
Some tasks change the amount: addition combines shares and subtraction removes or compares them. Other tasks change only its description: an equivalent fraction or a mixed-number conversion names the same quantity differently. Comparison asks how two quantities relate without altering either. Distinguishing these purposes helps you choose a method rather than following a rule blindly.
| Idea | Meaning or method | Example or check |
|---|---|---|
| Equal shares and parts | Use equal amounts of the same chosen whole | One roti for four children gives 1/4 roti each |
| Unit fractions | One of b equal parts is 1/b | For the same whole, 1/5 > 1/9 |
| Numerator and denominator | Count of pieces and the unit identifying their size | 5/6 is five copies of 1/6 |
| Measuring and number lines | Count equal lengths from zero | 4/5 is at eight tenths; infinitely many fractions lie between 0 and 1 |
| Beyond one and mixed numbers | Group b fractional units into each whole | 8/3 = 2 2/3; 3 3/4 = 15/4 |
| Equivalent fractions | Scale both numbers by the same factor | 1/2 = 2/4 = 4/8 |
| Lowest terms | Divide out common factors without changing the amount | 36/60 = 3/5 |
| Comparing and ordering | Use a common unit, then compare counts | 4/5 = 36/45 > 35/45 = 7/9 |
| Adding | Combine counts after making units match | 1/4 + 1/3 = 7/12 |
| Subtracting | Remove or compare counts after making units match | 3/4 − 2/3 = 1/12 |
| History and fraction words | People developed names, notation, and general methods | Brahmagupta described common-denominator arithmetic |
| Unit-fraction puzzles | Combine distinct-sized unit parts while checking the total | 1/2 + 1/3 + 1/6 = 1 |
See the same number in several ways
A useful revision task is to represent one quantity with a picture, a fraction, a mixed number when suitable, and a point on a number line. These representations should all agree. A shaded area counts pieces of a whole, a sharing story counts a person’s equal share, and a number line measures distance from zero. Moving between them tests understanding more deeply than recognising a symbol alone.
Problem
Four friends share five rotis equally. Represent each share as a fraction and a mixed number, and locate it.
- 1.Split every roti into quarters and give one quarter from each roti to each friend.
- 2.Each friend has five quarter-pieces, so the share is 5/4 of a roti.
- 3.Four quarters form one whole, leaving a quarter: 5/4 = 1 1/4.
- 4.On a number line divided into quarter-unit intervals, locate the fifth interval from zero. It is one quarter past 1.
- 5.Check the sharing total: four shares of 5/4 make twenty quarters, equal to all five original rotis.
The whole must stay consistent during a representation change. A half of one standard strip and two quarters of that same strip have equal lengths. A half of a different-sized strip may not match them. Likewise, differently shaped pieces can represent the same fraction if they cover equal amounts of equal wholes; identical outlines are not required.
Choose the method by identifying the task
Before calculating, say what the question asks: rename a quantity, put amounts in order, combine them, or find a difference. Then choose a helpful unit. A common denominator is useful for comparisons and arithmetic, while common factors simplify a single fraction. A mixed-number conversion groups fractional pieces into wholes or replaces wholes with pieces.
The following relationships collect the chapter’s main methods. In them, a and c are numerator counts, b is a positive denominator, k is a positive whole-number scaling factor, and w and r are a mixed number’s whole part and leftover numerator. For the subtraction relationship, the first quantity is at least the second. Each formula expresses reasoning you have already used with equal-sized pieces.
For unlike denominators, first rewrite the fractions using a common denominator. You do not need to memorise another unexplained formula: find the common unit, count the pieces, and simplify. The same preparation serves both comparison and arithmetic, but the final action depends on the question.
Problem
Simplify 16/20, compare it with 7/9, and find how much larger the greater fraction is.
- 1.Divide 16 and 20 by their common factor 4 to get 4/5.
- 2.For comparison use forty-fifths: 4/5 = 36/45 and 7/9 = 35/45.
- 3.Since 36 > 35, 16/20 is larger than 7/9.
- 4.The difference is 36/45 − 35/45 = 1/45. Simplification, comparison, and subtraction answer three different parts of the question.
Problem
Find 1 1/2 + 3/4.
- 1.Convert the mixed number: 1 1/2 = (1 × 2 + 1)/2 = 3/2.
- 2.Use quarters: 3/2 = 6/4.
- 3.Add 6/4 + 3/4 = 9/4.
- 4.Group nine quarters into two wholes and one quarter. The sum is 2 1/4.
A practical answer must fit its situation
A numerical result is only part of an answer. State whether it describes food per person, a remaining length, a volume, or a time difference. Check its measurement unit and compare it with the starting amounts. A sum of two positive quantities should exceed either addend, while removing part of a quantity should leave less than the starting amount.
Problem
A ribbon is 1 1/4 metres long. First 1/3 metre and then 1/2 metre are cut off. How much remains?
- 1.The original length is 5/4 metres.
- 2.Find the amount cut off: 1/3 + 1/2 = 2/6 + 3/6 = 5/6 metre.
- 3.Use twelfths to subtract: 5/4 = 15/12 and 5/6 = 10/12.
- 4.The remaining length is 15/12 − 10/12 = 5/12 metre.
- 5.Check by addition: 5/12 remaining + 10/12 removed = 15/12 = 1 1/4 metres, the original length.
Problem
Group A shares 3 litres among 4 children. Group B shares 6 litres among 8 children. Group C shares 7 litres among 10 children. Compare each child’s share.
- 1.The shares are 3/4, 6/8, and 7/10 litre.
- 2.Group B’s fraction simplifies to 3/4, so A and B receive the same amount.
- 3.In twentieths, 3/4 = 15/20 and 7/10 = 14/20.
- 4.Each child in A or B receives 1/20 litre more than each child in C. Scaling both supply and recipients preserves a share; changing them by different factors need not.
A larger denominator does not necessarily give a larger fraction. Equivalent fractions require the same multiplying or dividing factor in both numbers. Addition and subtraction require like-sized units; adding or subtracting the denominators changes the unit and gives an incorrect result.
Return to the whole through a puzzle
The distinct-unit-fraction puzzle links several chapter ideas. Comparison tells us which pieces are largest. Addition combines selected pieces, while subtraction finds the missing amount needed to reach one. Equivalent fractions check the sum exactly. The history of fraction notation and methods gives context, but your justification must still come from the sizes and counts of the pieces.
Problem
Why is 1/2 + 1/3 + 1/6 the only three-term sum of distinct unit fractions equal to one?
- 1.Without a half, all three terms are at most a third; distinct terms total less than three thirds, so cannot reach one.
- 2.After choosing a half, two different terms must total a half. Without a third, both would be at most a quarter, and distinct choices total less than two quarters.
- 3.The half and third are forced. Their sum is 5/6, leaving 1/6.
- 4.All terms are distinct and total one. Reordering them keeps the same solution.
Quiz
In 7/10, which number counts the pieces selected?
Which statement about one whole is correct?
Which fraction equals 2 2/3?
Which is 36/60 in lowest terms?
Which comparison correctly orders three quarters and seven tenths?
What is 2/3 + 1/5?
What is 3/4 − 2/3?
Why can two distinct unit fractions below one not sum to one?
Where is 9/4 on a number line?
What connects Brahmagupta’s addition and subtraction methods?
Practice Problems
- Draw the same whole split into halves, quarters, and sixths. Explain when pieces with different shapes can have equal fractional sizes.
- Three guavas of approximately equal weight total 1 kg. State each approximate weight. Explain why the equality is approximate.
- Use paper strips to show 1/2 = 2/4 = 4/8 and 1/3 = 2/6. Make a table of one through seven copies of 1/2.
- Locate 1/10, 3/10, 4/5, 3/2, and 9/5 on suitable number lines. Explain why your drawn points are not all the fractions between 0 and 1.
- Write 19/6 and 47/9 as mixed numbers, and 3 1/6 and 2 3/11 as fractions. Check one conversion by reversing it.
- Simplify 64/144 and 126/147. Find three fractions equivalent to each result and explain the unchanged value.
- Compare 4/9 with 3/7 and 9/4 with 5/2. Arrange 19/24, 5/6, and 7/12 in ascending order.
- Add and simplify 2/3 + 5/6, 3/5 + 5/8, and 9/2 + 5/4 + 7/6. Interpret one sum with a strip or a mixed number.
- Subtract 13/4 from 10/3 and find 5/6 − 4/9. Explain why the word from matters in the first instruction.
- Four children share three rotis. Write division, addition, and multiplication facts for the situation, then describe a doubled group with the same share.
- A border needs 1 metre of lace. Two lengths of 2/5 and 3/4 metre are available. Find the total, decide whether it is enough, and find the excess.
- A journey is 7/10 km long and 1/2 km is travelled by auto. Find the walking distance and check by addition. Compare times 10/3 and 13/4 minutes and find their difference.
- Explain why 1/3 + 1/4 is not 2/7, why 3/4 − 2/3 is not 1, and why adding one to both numbers in a fraction does not normally preserve it.
- Verify 19/24 = 1/2 + 1/6 + 1/8. Explain why two distinct proper unit fractions cannot total one, then construct a four-term sum equal to one.
- Ask for fraction words in a familiar language. Describe one historical development in notation and explain the shared idea behind Brahmagupta’s addition and subtraction methods.
Key Takeaways
• Fractions connect equal sharing, parts of a whole, measuring, and number-line positions. • The numerator counts fractional units and the denominator specifies their size. • Equivalent fractions and mixed-number conversions preserve the amount while changing its description. • Common fractional units make comparison, addition, and subtraction meaningful. • Check each answer with its whole, unit, representation, and practical meaning. • Fraction history and unit-fraction puzzles connect these methods with naming, reasoning, and discovery.
Previous · Lesson 8
Fractions Through History and Puzzles
Next
End of chapter