GSEB Class 6 Maths Solutions Chapter 7 Fractions InText Questions

Try These (Page 137)

 

Question 1. Show \( \frac{3}{5} \) on a number line.
Answer: To show \( \frac{3}{5} \) on a number line, we first draw a number line. We then divide the length between 0 and 1 into 5 equal parts. Each part will represent \( \frac{1}{5} \). The point A, which is the third mark from 0, will represent \( \frac{3}{5} \).
In simple words: First, draw a line and mark 0 and 1. Then, split the space between 0 and 1 into five equal parts. Count three parts from 0, and that spot is where \( \frac{3}{5} \) belongs.

Exam Tip: For representing fractions on a number line, ensure you divide the whole (usually 0 to 1) into the number of parts indicated by the denominator, then count the parts indicated by the numerator.

 

Question 2. Show \( \frac{1}{10}, \frac{0}{10}, \frac{5}{10} \) and \( \frac{10}{10} \) on a number line.
Answer: We draw a number line. Divide the length between 0 and 1 into 10 equal parts.
The point A represents \( \frac{0}{10} \).
The point B represents \( \frac{1}{10} \).
The point C represents \( \frac{5}{10} \).
The point D represents \( \frac{10}{10} \).
In simple words: Draw a line and mark from 0 to 1. Divide this space into ten equal sections. Mark each section \( \frac{1}{10} \), \( \frac{2}{10} \), and so on. Then, you can easily point out where \( \frac{0}{10} \), \( \frac{1}{10} \), \( \frac{5}{10} \), and \( \frac{10}{10} \) are located.

Exam Tip: When marking fractions on a number line, ensure your divisions are equally spaced. For fractions with 0 as the numerator, the point is at 0. For fractions where the numerator equals the denominator, the point is at 1.

 

Question 3. Can you show any other fraction between 0 and 1? Write five more fractions that you can show and depict them on the number line.
Answer: Yes, we can show any number of fractions that are greater than 0 and less than 1. Five other fractions, for example: \( \frac{2}{3}, \frac{6}{7}, \frac{1}{8}, \frac{4}{9} \) and \( \frac{4}{5} \), can be shown on a number line between 0 and 1. You should try to place these numbers on a number line by yourself as an exercise.
In simple words: Yes, you can always find more fractions between 0 and 1. Just pick some, like two-thirds or six-sevenths, and try to draw them on your number line.

Exam Tip: Remember that fractions between 0 and 1 always have a numerator smaller than their denominator. You can create countless examples by choosing different denominators.

 

Question 4. How many fractions lie between 0 and 1? Think, discuss and write your answer.
Answer: An infinite number of fractions lie between 0 and 1. This means there are endless possibilities to find fractions within this range.
In simple words: There are so many fractions between 0 and 1 that you can't even count them all; there's an endless amount.

Exam Tip: Understanding that there are infinite fractions between any two whole numbers is a key concept in understanding the density of rational numbers.

Try These (Page 138)

 

Question 1. Give a proper fraction:
(a) Whose numerator is 5 and denominator is 7.
(b) Whose denominator is 9 and numerator is 5.
(c) Whose numerator and denominator add up to 10. How many fractions of this kind can you make?
(d) Whose denominator is 4 more than the numerator. (Give any five. How many more can you make?)
Answer:
(a) If the numerator is 5 and the denominator is 7, the fraction is \( \frac{5}{7} \).
(b) If the denominator is 9 and the numerator is 5, the fraction is \( \frac{5}{9} \).
(c) Here are fractions whose numerator and denominator add up to 10:

These fractions are: \( \frac{0}{10}, \frac{1}{9}, \frac{2}{8}, \frac{3}{7} \) and \( \frac{4}{6} \). We can make 5 such fractions where the numerator is less than the denominator.
(d) There can be an infinite number of fractions whose denominator is 4 more than the numerator. Here are some examples: \( \frac{1}{5}, \frac{2}{6}, \frac{3}{7}, \frac{4}{8}, \frac{5}{9} \), etc.
In simple words: (a) It's just five over seven. (b) It's five over nine. (c) There are five fractions like \( \frac{0}{10} \) or \( \frac{1}{9} \) where the top and bottom numbers sum to ten. (d) You can make endless fractions where the bottom number is always four bigger than the top number, like \( \frac{1}{5} \) or \( \frac{2}{6} \).

Exam Tip: A proper fraction always has a numerator smaller than its denominator. This ensures the fraction's value is less than 1.

 

Question 2. A fraction is given. How will you decide, by just looking at it, whether, the fraction is (a) less than 1? (b) equal to 1?
Answer:
(a) If the numerator is less than the denominator, then the fraction is less than 1.
(b) If the numerator is equal to the denominator, then the fraction is equal to 1.
In simple words: (a) If the top number is smaller than the bottom number, the fraction is less than one. (b) If the top and bottom numbers are the same, the fraction is exactly equal to one.

Exam Tip: Quickly check if the numerator is smaller than, equal to, or greater than the denominator to understand if the fraction is proper, equal to one, or improper, respectively.

 

Question 3. Fill up using one of these: '>', '<' or '='
(a) \( \frac{1}{2} \) [] 1
(b) \( \frac{3}{5} \) [] 1
(c) 1 [] \( \frac{7}{8} \)
(d) \( \frac{4}{4} \) [] 1
(e) \( \frac{2005}{2005} \) [] 1
Answer:
(a) \( \frac{1}{2} < 1 \)
(b) \( \frac{3}{5} < 1 \)
(c) \( 1 > \frac{7}{8} \)
(d) \( \frac{4}{4} = 1 \)
(e) \( \frac{2005}{2005} = 1 \)
In simple words: (a) Half is smaller than one. (b) Three-fifths is smaller than one. (c) One is bigger than seven-eighths. (d) Four-fourths is the same as one whole. (e) Two thousand five over two thousand five is also the same as one whole.

Exam Tip: When the numerator is smaller than the denominator, the fraction is less than 1. When they are equal, the fraction is equal to 1.

Try These (Page 142)

 

Question 1. Are \( \frac{1}{3} \) and \( \frac{2}{7} \); \( \frac{2}{5} \) and \( \frac{2}{7} \); \( \frac{2}{9} \) and \( \frac{6}{27} \) equivalent? Give reason.
Answer:
(i) To check if \( \frac{1}{3} \) and \( \frac{2}{7} \) are equivalent, we cross-multiply:
\( 1 \times 7 = 7 \)
\( 3 \times 2 = 6 \)
Since \( 7 \neq 6 \), which means \( 1 \times 7 \neq 3 \times 2 \), these fractions are not equivalent.
(ii) To check if \( \frac{2}{5} \) and \( \frac{2}{7} \) are equivalent, we cross-multiply:
\( 2 \times 7 = 14 \)
\( 5 \times 2 = 10 \)
Since \( 14 \neq 10 \), which means \( 2 \times 7 \neq 5 \times 2 \), these fractions are not equivalent.
(iii) To check if \( \frac{2}{9} \) and \( \frac{6}{27} \) are equivalent, we cross-multiply:
\( 2 \times 27 = 54 \)
\( 9 \times 6 = 54 \)
Since \( 54 = 54 \), which means \( 2 \times 27 = 9 \times 6 \), these fractions are equivalent.
In simple words: You check if fractions are the same by multiplying across (numerator of one by denominator of the other). If the results are equal, they are equivalent. For \( \frac{1}{3} \) and \( \frac{2}{7} \), the results are 7 and 6, so they're not equal. For \( \frac{2}{5} \) and \( \frac{2}{7} \), the results are 14 and 10, also not equal. But for \( \frac{2}{9} \) and \( \frac{6}{27} \), both products are 54, so these are equivalent.

Exam Tip: Cross-multiplication is a reliable method to determine if two fractions are equivalent. If the cross-products are equal, the fractions are equivalent.

 

Question 2. Give an example of four equivalent fractions.
Answer: Four equivalent fractions can be formed by multiplying the numerator and the denominator of a fraction by the same non-zero integer. For example, starting with \( \frac{1}{4} \):
\( \frac{1}{4} \)
\( \frac{1 \times 2}{4 \times 2} = \frac{2}{8} \)
\( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \)
\( \frac{1 \times 4}{4 \times 4} = \frac{4}{16} \)
Thus, four equivalent fractions are \( \frac{1}{4}, \frac{2}{8}, \frac{3}{12}, \frac{4}{16} \).
In simple words: Equivalent fractions are just different ways to write the same amount. You can make them by multiplying both the top and bottom of a fraction by the same number, like how \( \frac{1}{4} \) is the same as \( \frac{2}{8} \) or \( \frac{3}{12} \).

Exam Tip: To find equivalent fractions, multiply or divide both the numerator and denominator by the same number. This process does not change the value of the fraction.

 

Question 3. Identify the fractions in each. Are these fractions equivalent?
Answer: Let's identify the fraction represented by each figure and then determine if they are equivalent.
(i) The figure shows 6 shaded parts out of a total of 8 parts. So, this figure represents the fraction \( \frac{6}{8} \). When simplified, \( \frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \).
(ii) The figure shows 9 shaded parts out of a total of 12 parts. So, this figure represents the fraction \( \frac{9}{12} \). When simplified, \( \frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4} \).
(iii) The figure shows 12 shaded parts out of a total of 16 parts. So, this figure represents the fraction \( \frac{12}{16} \). When simplified, \( \frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4} \).
(iv) The figure shows 15 shaded parts out of a total of 20 parts. So, this figure represents the fraction \( \frac{15}{20} \). When simplified, \( \frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4} \).
Since all the fractions simplify to \( \frac{3}{4} \), this means \( \frac{6}{8} = \frac{9}{12} = \frac{12}{16} = \frac{15}{20} \). Therefore, these figures represent equivalent fractions.
In simple words: We look at how many parts are colored out of the total parts in each picture. Then, we simplify each fraction. Because all of them simplify to \( \frac{3}{4} \), it means they all show the same amount, so they are equivalent.

Exam Tip: To check if fractions represented by diagrams are equivalent, always simplify each fraction to its simplest form. If their simplest forms are the same, the fractions are equivalent.

Try These (Page 143)

 

Question 1. Find five equivalent fractions of each of the following:
(i) \( \frac{2}{3} \)
(ii) \( \frac{1}{5} \)
(iii) \( \frac{3}{5} \)
(iv) \( \frac{5}{9} \)
Answer: To find equivalent fractions, we multiply both the numerator and the denominator by the same non-zero whole number.
(i) For \( \frac{2}{3} \), five equivalent fractions are:
\( \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \)
\( \frac{2 \times 3}{3 \times 3} = \frac{6}{9} \)
\( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)
\( \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \)
\( \frac{2 \times 6}{3 \times 6} = \frac{12}{18} \)
Thus, the equivalent fractions are \( \frac{4}{6}, \frac{6}{9}, \frac{8}{12}, \frac{10}{15}, \frac{12}{18} \).
(ii) For \( \frac{1}{5} \), five equivalent fractions are:
\( \frac{1 \times 2}{5 \times 2} = \frac{2}{10} \)
\( \frac{1 \times 3}{5 \times 3} = \frac{3}{15} \)
\( \frac{1 \times 4}{5 \times 4} = \frac{4}{20} \)
\( \frac{1 \times 5}{5 \times 5} = \frac{5}{25} \)
\( \frac{1 \times 6}{5 \times 6} = \frac{6}{30} \)
Thus, the equivalent fractions are \( \frac{2}{10}, \frac{3}{15}, \frac{4}{20}, \frac{5}{25}, \frac{6}{30} \).
(iii) For \( \frac{3}{5} \), five equivalent fractions are:
\( \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \)
\( \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \)
\( \frac{3 \times 4}{5 \times 4} = \frac{12}{20} \)
\( \frac{3 \times 5}{5 \times 5} = \frac{15}{25} \)
\( \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \)
Thus, the equivalent fractions are \( \frac{6}{10}, \frac{9}{15}, \frac{12}{20}, \frac{15}{25}, \frac{18}{30} \).
(iv) For \( \frac{5}{9} \), five equivalent fractions are:
\( \frac{5 \times 2}{9 \times 2} = \frac{10}{18} \)
\( \frac{5 \times 3}{9 \times 3} = \frac{15}{27} \)
\( \frac{5 \times 4}{9 \times 4} = \frac{20}{36} \)
\( \frac{5 \times 5}{9 \times 5} = \frac{25}{45} \)
\( \frac{5 \times 6}{9 \times 6} = \frac{30}{54} \)
Thus, the equivalent fractions are \( \frac{10}{18}, \frac{15}{27}, \frac{20}{36}, \frac{25}{45}, \frac{30}{54} \).
In simple words: To find fractions that are worth the same, just multiply both the top and bottom numbers by 2, then by 3, then by 4, and so on, for each fraction given. This creates new fractions that are equivalent.

Exam Tip: Remember to always multiply both the numerator and the denominator by the same number to generate equivalent fractions. Avoid using zero, as it would make the fraction undefined.

Try These (Page 146)

 

Question 1. Write the simplest form of:
(i) \( \frac{15}{75} \)
(ii) \( \frac{16}{72} \)
(iii) \( \frac{17}{51} \)
(iv) \( \frac{42}{28} \)
(v) \( \frac{80}{24} \)
Answer: To write a fraction in its simplest form, we divide both the numerator and the denominator by their Highest Common Factor (HCF).
(i) For \( \frac{15}{75} \):
Factors of 15 are: 1, 3, 5, 15.
Factors of 75 are: 1, 3, 5, 15, 25, 75.
The common factors are: 1, 3, 5, 15.
The HCF of 15 and 75 is 15.
Now, \( \frac{15}{75} = \frac{15 \div 15}{75 \div 15} = \frac{1}{5} \).
Thus, the simplest form of \( \frac{15}{75} \) is \( \frac{1}{5} \).
(ii) For \( \frac{16}{72} \):
Factors of 16 are: 1, 2, 4, 8, 16.
Factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
The common factors are: 1, 2, 4, 8.
The HCF of 16 and 72 is 8.
Now, \( \frac{16}{72} = \frac{16 \div 8}{72 \div 8} = \frac{2}{9} \).
Thus, the simplest form of \( \frac{16}{72} \) is \( \frac{2}{9} \).
(iii) For \( \frac{17}{51} \):
Factors of 17 are: 1, 17.
Factors of 51 are: 1, 3, 17, 51.
The common factor is 17.
The HCF of 17 and 51 is 17.
Now, \( \frac{17}{51} = \frac{17 \div 17}{51 \div 17} = \frac{1}{3} \).
Thus, the simplest form of \( \frac{17}{51} \) is \( \frac{1}{3} \).
(iv) For \( \frac{42}{28} \):
Factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42.
Factors of 28 are: 1, 2, 4, 7, 14, 28.
Common factors are: 1, 2, 7, 14.
The HCF of 42 and 28 is 14.
Now, \( \frac{42}{28} = \frac{42 \div 14}{28 \div 14} = \frac{3}{2} \).
Thus, the simplest form of \( \frac{42}{28} \) is \( \frac{3}{2} \).
(v) For \( \frac{80}{24} \):
Factors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80.
Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
Common factors are: 1, 2, 4, 8.
The HCF of 80 and 24 is 8.
Now, \( \frac{80}{24} = \frac{80 \div 8}{24 \div 8} = \frac{10}{3} \).
Thus, the simplest form of \( \frac{80}{24} \) is \( \frac{10}{3} \).
In simple words: To simplify a fraction, find the biggest number that divides both the top and bottom numbers evenly. Then, divide both by that number. Do this for each fraction until you can't divide them any more.

Exam Tip: Always look for the HCF (Highest Common Factor) of the numerator and denominator to simplify fractions in one step. If you can't find the HCF immediately, divide by any common factor until no more common factors (other than 1) exist.

 

Question 2. Is \( \frac{49}{64} \) in its simplest form?
Answer: To determine if \( \frac{49}{64} \) is in its simplest form, we check their common factors.
Factors of 49 are: 1, 7, 49.
Factors of 64 are: 1, 2, 4, 8, 16, 32, 64.
The only common factor between 49 and 64 is 1. Since 49 and 64 share no common factor other than 1, they are coprime numbers. Therefore, the fraction \( \frac{49}{64} \) is in its simplest form.
In simple words: To check if a fraction is as simple as it can get, find all the numbers that can divide the top number and all the numbers that can divide the bottom number. If the only number they both share is 1, then the fraction is already in its simplest form. For \( \frac{49}{64} \), only 1 is a common divisor, so it's simple.

Exam Tip: A fraction is in its simplest form when the only common factor between its numerator and denominator is 1. If you find any other common factor, the fraction can be simplified further.

Try These (Page 148)

 

Question 1. You get one-fifth of a bottle of juice and your sister gets one-third of a bottle of juice. Who gets more?
Answer: Let's compare the fractions \( \frac{1}{5} \) and \( \frac{1}{3} \).
We can visualize this by dividing a rectangle into equal parts.
For \( \frac{1}{5} \), imagine a rectangle divided into 5 equal parts. If you shade one of these parts, that represents one-fifth.

For \( \frac{1}{3} \), imagine the same sized rectangle divided into 3 equal parts. If you shade one of these parts, that represents one-third.

By comparing the shaded areas, it's clear that one-third is a larger portion than one-fifth. Thus, your sister gets more juice.
In simple words: To see who gets more, compare \( \frac{1}{5} \) and \( \frac{1}{3} \). Imagine cutting a pie into 5 slices and another pie into 3 slices. A single slice from the 3-slice pie (one-third) will be bigger than a single slice from the 5-slice pie (one-fifth). So, your sister gets more.

Exam Tip: When comparing unit fractions (fractions with a numerator of 1), the fraction with the smaller denominator is always larger because the whole is divided into fewer, larger parts.

Try These (Page 149)

 

Question 1.
(a) Which is the larger fraction?
(i) \( \frac{7}{10} \) or \( \frac{8}{10} \)
(ii) \( \frac{11}{24} \) or \( \frac{13}{24} \)
(iii) \( \frac{17}{102} \) or \( \frac{12}{102} \)
(b) Why are these comparisons easy to make?
Answer:
(a) (i) To compare \( \frac{7}{10} \) and \( \frac{8}{10} \): Since the denominators are the same (10), we compare the numerators. \( 7 < 8 \), therefore \( \frac{7}{10} < \frac{8}{10} \). The larger fraction is \( \frac{8}{10} \).
(ii) To compare \( \frac{11}{24} \) and \( \frac{13}{24} \): Since the denominators are the same (24), we compare the numerators. \( 11 < 13 \), therefore \( \frac{11}{24} < \frac{13}{24} \). The larger fraction is \( \frac{13}{24} \).
(iii) To compare \( \frac{17}{102} \) and \( \frac{12}{102} \): Since the denominators are the same (102), we compare the numerators. \( 17 > 12 \), therefore \( \frac{17}{102} > \frac{12}{102} \). The larger fraction is \( \frac{17}{102} \).
(b) These comparisons are easy to make because these are "like fractions." Like fractions have the same denominator, which simplifies the comparison to just looking at their numerators.
In simple words: (a) To find the bigger fraction, just look at the top numbers because the bottom numbers are all the same. So, \( \frac{8}{10} \) is bigger than \( \frac{7}{10} \), \( \frac{13}{24} \) is bigger than \( \frac{11}{24} \), and \( \frac{17}{102} \) is bigger than \( \frac{12}{102} \). (b) These are easy to compare because all the fractions share the same bottom number.

Exam Tip: Comparing like fractions (fractions with the same denominator) is straightforward: the fraction with the larger numerator is the greater fraction.

 

Question 2. We can write the like fractions in ascending or in descending order according to the order of their numerators.
(a) \( \frac{1}{8}, \frac{5}{8}, \frac{3}{8} \)
(b) \( \frac{1}{5}, \frac{11}{5}, \frac{4}{5}, \frac{3}{5}, \frac{7}{5} \)
(c) \( \frac{1}{7}, \frac{3}{7}, \frac{13}{7}, \frac{11}{7}, \frac{7}{7} \)
Answer: When fractions have the same denominator (like fractions), we can arrange them by simply ordering their numerators.
(a) Given fractions: \( \frac{1}{8}, \frac{5}{8}, \frac{3}{8} \). The numerators are 1, 5, and 3.
In ascending order of numerators: 1, 3, 5.
So, ascending order of fractions: \( \frac{1}{8}, \frac{3}{8}, \frac{5}{8} \).
In descending order of numerators: 5, 3, 1.
So, descending order of fractions: \( \frac{5}{8}, \frac{3}{8}, \frac{1}{8} \).
(b) Given fractions: \( \frac{1}{5}, \frac{11}{5}, \frac{4}{5}, \frac{3}{5}, \frac{7}{5} \). The numerators are 1, 11, 4, 3, and 7.
In ascending order of numerators: 1, 3, 4, 7, 11.
So, ascending order of fractions: \( \frac{1}{5}, \frac{3}{5}, \frac{4}{5}, \frac{7}{5}, \frac{11}{5} \).
In descending order of numerators: 11, 7, 4, 3, 1.
So, descending order of fractions: \( \frac{11}{5}, \frac{7}{5}, \frac{4}{5}, \frac{3}{5}, \frac{1}{5} \).
(c) Given fractions: \( \frac{1}{7}, \frac{3}{7}, \frac{13}{7}, \frac{11}{7}, \frac{7}{7} \). The numerators are 1, 3, 13, 11, and 7.
In ascending order of numerators: 1, 3, 7, 11, 13.
So, ascending order of fractions: \( \frac{1}{7}, \frac{3}{7}, \frac{7}{7}, \frac{11}{7}, \frac{13}{7} \).
In descending order of numerators: 13, 11, 7, 3, 1.
So, descending order of fractions: \( \frac{13}{7}, \frac{11}{7}, \frac{7}{7}, \frac{3}{7}, \frac{1}{7} \).
In simple words: When the bottom number of fractions is the same, putting them in order is easy. Just look at the top numbers and sort them from smallest to largest for ascending order, or largest to smallest for descending order.

Exam Tip: Always double-check that the fractions are 'like fractions' (have the same denominator) before sorting them solely by their numerators. If they are not, convert them to like fractions first.

Try These (Page 151)

 

Question 1. Arrange the following in ascending and descending order:
(a) \( \frac{1}{12}, \frac{1}{23}, \frac{1}{5}, \frac{1}{7}, \frac{1}{50}, \frac{1}{9}, \frac{1}{17} \)
(b) \( \frac{3}{7}, \frac{3}{11}, \frac{3}{5}, \frac{3}{2}, \frac{3}{13}, \frac{3}{4}, \frac{3}{17} \)
(c) Write 3 more similar examples and arrange them in ascending and descending order.
Answer: We know that for 'unlike' fractions having the same numerator, the greater the value of the denominator, the smaller the value of the fractional number.
(a) Given fractions: \( \frac{1}{12}, \frac{1}{23}, \frac{1}{5}, \frac{1}{7}, \frac{1}{50}, \frac{1}{9}, \frac{1}{17} \). All have a numerator of 1. The denominators are 12, 23, 5, 7, 50, 9, 17.
In descending order of denominators: 50, 23, 17, 12, 9, 7, 5.
So, ascending order of fractions (smallest denominator gives largest fraction):
\( \frac{1}{50}, \frac{1}{23}, \frac{1}{17}, \frac{1}{12}, \frac{1}{9}, \frac{1}{7}, \frac{1}{5} \).
And descending order of fractions:
\( \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \frac{1}{12}, \frac{1}{17}, \frac{1}{23}, \frac{1}{50} \).
(b) Given fractions: \( \frac{3}{7}, \frac{3}{11}, \frac{3}{5}, \frac{3}{2}, \frac{3}{13}, \frac{3}{4}, \frac{3}{17} \). All have a numerator of 3. The denominators are 7, 11, 5, 2, 13, 4, 17.
In descending order of denominators: 17, 13, 11, 7, 5, 4, 2.
So, ascending order of fractions:
\( \frac{3}{17}, \frac{3}{13}, \frac{3}{11}, \frac{3}{7}, \frac{3}{5}, \frac{3}{4}, \frac{3}{2} \).
And descending order of fractions:
\( \frac{3}{2}, \frac{3}{4}, \frac{3}{5}, \frac{3}{7}, \frac{3}{11}, \frac{3}{13}, \frac{3}{17} \).
(c) Three more examples of unlike fractions with the same numerator are:
(i) \( \frac{2}{13}, \frac{2}{25}, \frac{2}{6}, \frac{2}{8}, \frac{2}{10}, \frac{2}{17} \)
Ascending order: \( \frac{2}{25}, \frac{2}{17}, \frac{2}{13}, \frac{2}{10}, \frac{2}{8}, \frac{2}{6} \).
Descending order: \( \frac{2}{6}, \frac{2}{8}, \frac{2}{10}, \frac{2}{13}, \frac{2}{17}, \frac{2}{25} \).
(ii) \( \frac{5}{6}, \frac{5}{17}, \frac{5}{14}, \frac{5}{7}, \frac{5}{12}, \frac{5}{8}, \frac{5}{11} \)
Ascending order: \( \frac{5}{17}, \frac{5}{14}, \frac{5}{12}, \frac{5}{11}, \frac{5}{8}, \frac{5}{7}, \frac{5}{6} \).
Descending order: \( \frac{5}{6}, \frac{5}{7}, \frac{5}{8}, \frac{5}{11}, \frac{5}{12}, \frac{5}{14}, \frac{5}{17} \).
(iii) \( \frac{4}{11}, \frac{4}{13}, \frac{4}{5}, \frac{4}{7}, \frac{4}{19}, \frac{4}{15} \)
Ascending order: \( \frac{4}{19}, \frac{4}{15}, \frac{4}{13}, \frac{4}{11}, \frac{4}{7}, \frac{4}{5} \).
Descending order: \( \frac{4}{5}, \frac{4}{7}, \frac{4}{11}, \frac{4}{13}, \frac{4}{15}, \frac{4}{19} \).
In simple words: When the top numbers of fractions are the same, the fraction with the biggest bottom number is actually the smallest overall. So, to order them from smallest to largest, you list them from the largest bottom number to the smallest bottom number. For reverse order, you do the opposite.

Exam Tip: For fractions with the same numerator, the one with the largest denominator has the smallest value. This rule is crucial for ordering unlike fractions with common numerators.

Try These (Page 155)

 

Question 1. My mother divided an apple into 4 equal parts. She gave me two parts and my brother one part. How many apples did she give to both of us together?
Answer: The total number of equal parts of the apple is 4. My mother gave me two parts, which is \( \frac{2}{4} \) of the apple. My brother received one part, which is \( \frac{1}{4} \) of the apple. To find how many apples she gave us together, we add these fractions:
\( \frac{2}{4} + \frac{1}{4} = \frac{2+1}{4} = \frac{3}{4} \)
So, she gave \( \frac{3}{4} \) of the apple to both of us together.
In simple words: Your mother cut an apple into four pieces. You got two pieces, and your brother got one. Together, you got three pieces, which is three-fourths of the apple.

Exam Tip: When adding fractions with the same denominator, simply add the numerators and keep the denominator the same. Always simplify the final fraction if possible.

 

Question 2. Mother asked Neelu and her brother to pick stones from the wheat. Neelu picked one-fourth of the total stones and her brother also picked up one-fourth of the stones. What fraction of the stones did both pick up together?
Answer: Neelu picked one-fourth of the total stones, i.e., \( \frac{1}{4} \) stones from the wheat. Her brother also picked up one-fourth of the stones, i.e., \( \frac{1}{4} \) stones. To find the total fraction of stones picked up together, we add the fractions:
\( \frac{1}{4} + \frac{1}{4} = \frac{1+1}{4} = \frac{2}{4} \)
Simplifying \( \frac{2}{4} \): The HCF of 2 and 4 is 2. So, \( \frac{2 \div 2}{4 \div 2} = \frac{1}{2} \).
Therefore, both Neelu and her brother picked up \( \frac{1}{2} \) of the total stones together.
In simple words: Neelu picked one-fourth of the stones, and her brother picked another one-fourth. When you add those together, they picked two-fourths, which is the same as half of all the stones.

Exam Tip: Remember to simplify fractions to their lowest terms whenever possible. This makes the answer clearer and often easier to understand.

 

Question 3. Sohan was putting covers on his notebooks. He put one-fourth of the covers on Monday. He put another one-fourth on Tuesday and the remaining on Wednesday. What fraction of the covers did he put on Wednesday?
Answer: Sohan put one-fourth of the covers on Monday, which is \( \frac{1}{4} \). He put another one-fourth of the covers on Tuesday, which is \( \frac{1}{4} \).
First, let's find the total fraction of covers put on Monday and Tuesday:
\( \frac{1}{4} + \frac{1}{4} = \frac{1+1}{4} = \frac{2}{4} \)
Simplifying \( \frac{2}{4} \): The HCF of 2 and 4 is 2. So, \( \frac{2 \div 2}{4 \div 2} = \frac{1}{2} \).
So, Sohan covered \( \frac{1}{2} \) of his notebooks by the end of Tuesday.
The remaining covers were put on Wednesday. If the total covers are represented by 1 (or \( \frac{4}{4} \)), then the fraction put on Wednesday is:
\( 1 - \frac{1}{2} = \frac{2}{2} - \frac{1}{2} = \frac{2-1}{2} = \frac{1}{2} \).
Therefore, Sohan put \( \frac{1}{2} \) of the covers on Wednesday.
In simple words: Sohan covered one-fourth of his books on Monday and another one-fourth on Tuesday. That's two-fourths, or half of his books. If he had a whole set of books (1) and covered half, the other half (which is \( \frac{1}{2} \)) must have been covered on Wednesday.

Exam Tip: When dealing with "remaining" quantities, always consider the total as 1 (or a fraction where the numerator equals the denominator, e.g., \( \frac{4}{4} \)) and subtract the portions already used.

Try These (Page 156)

 

Question 1. Add with the help of a diagram.
(i) \( \frac{1}{8} + \frac{1}{8} \)
(ii) \( \frac{2}{5} + \frac{3}{5} \)
(iii) \( \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \)
Answer: We can add these fractions using visual diagrams.
(i) For \( \frac{1}{8} + \frac{1}{8} \):
Method I: Look at the figure. It is divided into 8 equal parts.Its shaded part represents \( \frac{2}{8} \).
So, \( \frac{1}{8} + \frac{1}{8} = \frac{1+1}{8} = \frac{2}{8} \). This can be simplified to \( \frac{1}{4} \).
Method II: We can also represent the above sum in the following manner:
So, \( \frac{1}{8} + \frac{1}{8} = \frac{1+1}{8} = \frac{2}{8} \).
(ii) For \( \frac{2}{5} + \frac{3}{5} \):
Thus, \( \frac{2}{5} + \frac{3}{5} = \frac{2+3}{5} = \frac{5}{5} = 1 \).
(iii) For \( \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \):
Thus, \( \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1+1+1}{6} = \frac{3}{6} = \frac{1}{2} \).
In simple words: For each problem, draw a shape and divide it into parts based on the bottom number. Color in the number of parts from the top number. Then, combine the colored parts to see the total sum. Simplify the answer if you can.

Exam Tip: Visualizing fractions with diagrams helps to understand addition, especially for like fractions where you are just combining equal-sized parts.

 

Question 2. Add \( \frac{1}{12} + \frac{1}{12} \). How will we show this pictorially and by using paper folding?
Answer: To add \( \frac{1}{12} + \frac{1}{12} \):
\( \frac{1}{12} + \frac{1}{12} = \frac{1+1}{12} = \frac{2}{12} \)
Simplifying \( \frac{2}{12} \): The HCF of 2 and 12 is 2. So, \( \frac{2 \div 2}{12 \div 2} = \frac{1}{6} \).
So, \( \frac{1}{12} + \frac{1}{12} = \frac{1}{6} \).
To show this pictorially, we have:
Using paper folding (the activity): You should try this yourself by folding paper into 12 equal parts and shading.
In simple words: Adding \( \frac{1}{12} \) and \( \frac{1}{12} \) gives \( \frac{2}{12} \), which simplifies to \( \frac{1}{6} \). You can show this by drawing a box divided into 12 parts, coloring one part, then another, to see two colored parts. Or, fold a piece of paper into 12 equal strips, mark two, and then see how many bigger sections that makes.

Exam Tip: Paper folding is an excellent hands-on method to understand fraction addition. It visually reinforces the concept of combining parts of a whole.

 

Question 3. Make 5 more examples of problems given in 1 and 2 above. Solve them with your friends.
Answer: You should try to do this yourself as an activity to practice your fraction addition skills with diagrams.
In simple words: Create five new addition problems like the ones you just did, and then work them out with your friends using pictures or paper folding.

Exam Tip: Practicing with self-generated examples helps solidify understanding. Collaboration with peers can also offer new perspectives and methods for solving problems.

Try These (Page 157)

Question 1. Find the difference between \( \frac{7}{8} \) and \( \frac{3}{8} \).
Answer: To find the difference between \( \frac{7}{8} \) and \( \frac{3}{8} \), we subtract the second fraction from the first:
\( \frac{7}{8} - \frac{3}{8} = \frac{7-3}{8} = \frac{4}{8} \)
This fraction \( \frac{4}{8} \) can be simplified by dividing both the numerator and the denominator by their HCF, which is 4:
\( \frac{4 \div 4}{8 \div 4} = \frac{1}{2} \)
So, the difference between \( \frac{7}{8} \) and \( \frac{3}{8} \) is \( \frac{1}{2} \).
In simple words: When you subtract fractions with the same bottom number, you simply subtract the top numbers and keep the bottom number as it is. Then, make sure your final answer is as simple as possible.

Exam Tip: Similar to addition, when subtracting like fractions, subtract only the numerators and keep the common denominator. Always simplify the resulting fraction to its lowest terms.

 

Question 2. Mother made a good Patti in a round shape. She divided it into 5 parts. Seema ate one piece from it. If I eat another piece then how much would be left?
Answer: The total number of equal parts in the gud patti was 5. Seema ate one piece, which is \( \frac{1}{5} \) of the patti. You also ate one piece, which is another \( \frac{1}{5} \).
First, calculate the total fraction of gud patti eaten by both Seema and you:
Fraction eaten \( = \frac{1}{5} + \frac{1}{5} = \frac{1+1}{5} = \frac{2}{5} \)
The total gud patti can be represented as 1 whole, or \( \frac{5}{5} \). To find out how much was left, subtract the eaten portion from the total:
Fraction left over \( = 1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{5-2}{5} = \frac{3}{5} \)
Therefore, \( \frac{3}{5} \) of the gud patti would be left.
In simple words: The patti had 5 equal parts. Seema ate 1 part, and you ate 1 part. So, 2 parts were eaten in total. This means 3 parts are left out of 5, or \( \frac{3}{5} \) of the patti.

Exam Tip: For problems involving 'parts eaten' or 'parts removed' from a whole, first sum up the parts that are taken away. Then, subtract this total from the representation of the whole (e.g., \( \frac{5}{5} \) for 5 parts, or 1 for the whole). Always clearly state the total and the amount remaining.

 

Question 3. My elder sister divided the watermelon into 16 parts. I ate 7 out of them. My friend ate 4. How much did we eat altogether? How much more of the watermelon did I eat than my friend? What portion of the watermelon remained?
Answer: The watermelon was divided into 16 equal parts.
I ate \( \frac{7}{16} \) of the watermelon.
My friend ate \( \frac{4}{16} \) of the watermelon.

1. To find out how much we ate altogether, we add our portions:
Fraction eaten altogether \( = \frac{7}{16} + \frac{4}{16} = \frac{7+4}{16} = \frac{11}{16} \).
So, we ate \( \frac{11}{16} \) of the watermelon altogether.

2. To find out how much more I ate than my friend, we subtract my friend's portion from mine:
Difference in portions \( = \frac{7}{16} - \frac{4}{16} = \frac{7-4}{16} = \frac{3}{16} \).
So, I ate \( \frac{3}{16} \) more of the watermelon than my friend.

3. To find the portion that remained, we subtract the total eaten from the whole watermelon (which is \( \frac{16}{16} \)):
Fraction remaining \( = 1 - \frac{11}{16} = \frac{16}{16} - \frac{11}{16} = \frac{16-11}{16} = \frac{5}{16} \).
So, \( \frac{5}{16} \) of the watermelon remained.
In simple words: The watermelon was cut into 16 pieces. You ate 7 pieces, and your friend ate 4. So, you both ate 11 pieces together. You ate 3 more pieces than your friend. Since 11 pieces were eaten, 5 pieces were left over from the original 16.

Exam Tip: Break down complex word problems into smaller, manageable parts. Address each question separately, and clearly show your fraction addition and subtraction steps. Remember that 'the whole' can be represented as \( \frac{16}{16} \) or 1.

 

Question 4. Make five problems of this type and solve them with your friends.
Answer: This is an activity-based question that you should complete yourself. Create five new word problems involving adding or subtracting fractions, similar to the previous questions, and then work through the solutions with your friends. This practice will help you develop a deeper understanding of fractional operations in real-world contexts.
In simple words: This question asks you to make up five new fraction problems, like the ones you just did, and then solve them with your friends. It's a way to practice what you've learned.

Exam Tip: Designing your own problems helps you understand the underlying structure and application of fraction operations. It's a valuable way to reinforce learning and prepare for varied problem types.

Try These (Page 159)

Question 1. Add \( \frac{2}{5} + \frac{3}{7} \).
Answer: To add \( \frac{2}{5} \) and \( \frac{3}{7} \), we first need to find a common denominator. The Least Common Multiple (LCM) of 5 and 7 is 35.
Convert each fraction to have a denominator of 35:
For \( \frac{2}{5} \): multiply both numerator and denominator by 7.
\( \frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35} \)
For \( \frac{3}{7} \): multiply both numerator and denominator by 5.
\( \frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35} \)
Now, add the equivalent fractions:
\( \frac{14}{35} + \frac{15}{35} = \frac{14+15}{35} = \frac{29}{35} \)
So, \( \frac{2}{5} + \frac{3}{7} = \frac{29}{35} \).
In simple words: To add fractions with different bottom numbers, you must first find a common bottom number. You do this by finding the smallest number that both denominators can divide into. Then, change both fractions to use this new bottom number, and finally, add the top numbers.

Exam Tip: When adding or subtracting unlike fractions, always find the Least Common Multiple (LCM) of the denominators to create equivalent fractions with a common denominator. This makes the calculation accurate and simpler.

 

Question 2. Subtract \( \frac{2}{5} \) from \( \frac{5}{7} \).
Answer: To subtract \( \frac{2}{5} \) from \( \frac{5}{7} \), we first need to find a common denominator. The Least Common Multiple (LCM) of 7 and 5 is 35.
Convert each fraction to have a denominator of 35:
For \( \frac{5}{7} \): multiply both numerator and denominator by 5.
\( \frac{5}{7} = \frac{5 \times 5}{7 \times 5} = \frac{25}{35} \)
For \( \frac{2}{5} \): multiply both numerator and denominator by 7.
\( \frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35} \)
Now, subtract the equivalent fractions:
\( \frac{25}{35} - \frac{14}{35} = \frac{25-14}{35} = \frac{11}{35} \)
So, \( \frac{5}{7} - \frac{2}{5} = \frac{11}{35} \).
In simple words: To subtract fractions that have different bottom numbers, first find a common bottom number, usually the smallest one both denominators can divide into. Then, change both fractions to use this common bottom number and subtract the top numbers.

Exam Tip: When subtracting fractions, ensure you correctly identify which fraction is being subtracted from which. "Subtract A from B" means B - A. Always find the LCM for unlike fractions before performing subtraction.