Samacheer Kalvi Class 6 Maths Solutions Term 3 Chapter 1 Fractions Exercise 1.2

Miscellaneous Practice Problems

 

Question 1. Sankari purchased \( 2\frac{1}{2} \) m cloth to stitch a long skirt and \( 1\frac{3}{4} \) m cloth to stitch blouse. If the cost is Rs. 120 per metre then find the cost of cloth purchased by her.
Answer: First, we find the total length of cloth Sankari bought. She bought \( 2\frac{1}{2} \) m for a skirt and \( 1\frac{3}{4} \) m for a blouse. We add these mixed fractions:
Total cloth purchased \( = 2\frac{1}{2} + 1\frac{3}{4} \) m
Convert to improper fractions:
\( = \left( \frac{5}{2} + \frac{7}{4} \right) \) m
Find a common denominator, which is 4:
\( = \left( \frac{10}{4} + \frac{7}{4} \right) \) m
Add the numerators:
\( = \frac{10+7}{4} \) m
\( = \frac{17}{4} \) m
Now, we calculate the total cost. The cost of 1 metre of cloth is Rs. 120.
Total cost \( = 120 \times \frac{17}{4} \)
We can simplify by dividing 120 by 4, which is 30:
\( = 30 \times 17 \)
\( = \text{Rs. } 510 \)
So, Sankari spent Rs. 510 in total. This helps us understand how mixed numbers are used in everyday shopping calculations.
In simple words: Sankari bought cloth for a skirt and a blouse. We add the lengths to get the total cloth. Then, we multiply the total cloth by the price per metre to find the total cost.

🎯 Exam Tip: Always convert mixed fractions to improper fractions before performing addition or subtraction, as it simplifies calculations. Remember to specify units (m, Rs.) in your answer.

 

Question 2. From his office, a person wants to reach his house on foot which is at a distance of \( 5\frac{3}{4} \) km. If he had walked \( 2\frac{1}{2} \) km, how much distance still he has to walk to reach his house?
Answer: We need to find the remaining distance to walk. The total distance to the house is \( 5\frac{3}{4} \) km, and the person has already walked \( 2\frac{1}{2} \) km. We subtract the walked distance from the total distance:
Total distance \( = 5\frac{3}{4} \) km
Distance walked \( = 2\frac{1}{2} \) km
Distance to be walked \( = 5\frac{3}{4} - 2\frac{1}{2} \) km
Convert mixed fractions to improper fractions:
\( = \left( \frac{23}{4} - \frac{5}{2} \right) \) km
Find a common denominator, which is 4:
\( = \left( \frac{23}{4} - \frac{10}{4} \right) \) km
Subtract the numerators:
\( = \frac{23-10}{4} \) km
\( = \frac{13}{4} \) km
Convert back to a mixed fraction:
\( = 3\frac{1}{4} \) km
Therefore, the person still has to walk \( 3\frac{1}{4} \) km. This calculation helps us understand distances and journeys in real life.
In simple words: To find how much distance is left, subtract the distance already walked from the total distance to the house.

🎯 Exam Tip: When subtracting mixed fractions, ensure you convert them to improper fractions first and then find a common denominator for accurate results.

 

Question 3. Which is smaller? The difference between \( 2\frac{1}{2} \) and \( 3\frac{2}{3} \) or the sum of \( 1\frac{1}{2} \) and \( 2\frac{1}{4} \).
Answer: We need to calculate two values and then compare them.

First, let's find the difference between \( 3\frac{2}{3} \) and \( 2\frac{1}{2} \). We always subtract the smaller number from the larger one for the difference:
Difference \( = 3\frac{2}{3} - 2\frac{1}{2} \)
Convert to improper fractions:
\( = \frac{11}{3} - \frac{5}{2} \)
Find a common denominator, which is 6:
\( = \frac{(11 \times 2) - (3 \times 5)}{3 \times 2} \)
\( = \frac{22 - 15}{6} \)
\( = \frac{7}{6} \)

Next, let's find the sum of \( 1\frac{1}{2} \) and \( 2\frac{1}{4} \):
Sum \( = 1\frac{1}{2} + 2\frac{1}{4} \)
Convert to improper fractions:
\( = \frac{3}{2} + \frac{9}{4} \)
Find a common denominator, which is 4:
\( = \frac{(3 \times 2) + 9}{4} \)
\( = \frac{6 + 9}{4} \)
\( = \frac{15}{4} \)

Now we compare \( \frac{7}{6} \) and \( \frac{15}{4} \). To compare, we find a common denominator, which is 12:
\( \frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12} \)
\( \frac{15}{4} = \frac{15 \times 3}{4 \times 3} = \frac{45}{12} \)
Since \( \frac{14}{12} < \frac{45}{12} \), the difference \( \frac{7}{6} \) is smaller than the sum \( \frac{15}{4} \). Comparing fractions with a common denominator makes it easy to see which value is smaller or larger.
In simple words: First, calculate the difference between the first two numbers. Then, calculate the sum of the next two numbers. Finally, compare these two results to see which one is smaller.

🎯 Exam Tip: When comparing fractions, always convert them to equivalent fractions with a common denominator. This makes direct comparison of their numerators possible.

 

Question 4. Mangai bought \( 6\frac{3}{4} \) kg of apples. If Kalai bought \( 1\frac{1}{2} \) times a Mangai bought, then how many kilograms of apples did Kalai buy?
Answer: Mangai bought \( 6\frac{3}{4} \) kg of apples. Kalai bought \( 1\frac{1}{2} \) times the amount Mangai bought. To find out how many kilograms Kalai bought, we multiply Mangai's amount by \( 1\frac{1}{2} \).
Apples bought by Mangai \( = 6\frac{3}{4} \) kg
Kalai bought \( 1\frac{1}{2} \) times Mangai's amount.
Apples bought by Kalai \( = 6\frac{3}{4} \times 1\frac{1}{2} \) Kg
Convert mixed fractions to improper fractions:
\( = \frac{27}{4} \times \frac{3}{2} \) Kg
Multiply the numerators and the denominators:
\( = \frac{27 \times 3}{4 \times 2} \) Kg
\( = \frac{81}{8} \) Kg
Convert the improper fraction back to a mixed fraction:
\( = 10\frac{1}{8} \) Kg
So, Kalai bought \( 10\frac{1}{8} \) kg of apples. Understanding how to multiply fractions is very useful for scaling quantities, like in recipes or shopping.
In simple words: Kalai bought apples that were one and a half times the amount Mangai bought. So, we multiply Mangai's apple weight by one and a half to get Kalai's apple weight.

🎯 Exam Tip: "Times" in a word problem always indicates multiplication. Ensure all mixed fractions are converted to improper fractions before multiplying.

 

Question 5. The length of the staircase is \( 5\frac{1}{2} \) m. If one step is set at \( \frac{1}{4} \) m, then how many steps will be there in the staircase?
Answer: We need to find the number of steps in the staircase. The total length of the staircase is \( 5\frac{1}{2} \) m, and each step has a length of \( \frac{1}{4} \) m. To find the number of steps, we divide the total length by the length of one step.
Total length of the staircase \( = 5\frac{1}{2} \) m
Length of each step \( = \frac{1}{4} \) m
Number of steps in the staircase \( = \frac{\text{Total length}}{\text{Length of each step}} \)
\( = 5\frac{1}{2} \div \frac{1}{4} \)
Convert the mixed fraction to an improper fraction:
\( = \frac{11}{2} \div \frac{1}{4} \)
To divide by a fraction, we multiply by its reciprocal:
\( = \frac{11}{2} \times \frac{4}{1} \)
\( = \frac{11 \times 4}{2 \times 1} \)
Simplify by dividing 4 by 2:
\( = \frac{11 \times 2}{1 \times 1} \)
\( = 22 \) steps
There will be 22 steps in the staircase. Dividing fractions is essential for problems involving distributing a total quantity into smaller equal parts.
In simple words: Divide the total length of the staircase by the length of one step to find out how many steps there are.

🎯 Exam Tip: When dividing by a fraction, remember to multiply by its reciprocal (flip the second fraction). Always convert mixed numbers to improper fractions before division.

Challenge Problems

 

Question 6. By using the following clues, find who am I?
(i) Each of my numerator and denominator is a single-digit number.
(ii) The sum of my numerator and denominator is a multiple of 3.
(iii) The product of my numerator and denominator is a multiple of 4.
Answer: Let the fraction be \( \frac{a}{b} \).
(i) Both \( a \) and \( b \) are single-digit numbers. This means \( a, b \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). The problem mentions "proper fractions" later, implying \( a < b \).
(ii) The sum of my numerator and denominator is a multiple of 3. So, \( a+b \) is a multiple of 3. Possible sums: 3, 6, 9, 12, 15.
Let's list pairs \( (a,b) \) with \( a < b \) where \( a,b \) are single digits and \( a+b \) is a multiple of 3:
If \( a+b=3 \): (1,2)
If \( a+b=6 \): (1,5), (2,4)
If \( a+b=9 \): (1,8), (2,7), (3,6), (4,5)
If \( a+b=12 \): (3,9), (4,8), (5,7)
If \( a+b=15 \): (6,9), (7,8)
Possible proper fractions: \( \frac{1}{2}, \frac{1}{5}, \frac{2}{4}, \frac{1}{8}, \frac{2}{7}, \frac{3}{6}, \frac{4}{5}, \frac{3}{9}, \frac{4}{8}, \frac{5}{7}, \frac{6}{9}, \frac{7}{8} \). (The source uses \( \frac{4}{5} \) and \( \frac{5}{7} \), \( \frac{7}{8} \) even though they are proper, but it also includes \( \frac{6}{9} \). We list all possible proper fractions where \( a < b \)).

(iii) The product of my numerator and denominator is a multiple of 4. So, \( a \times b \) is a multiple of 4.
Let's check the fractions from the list above:
For \( \frac{1}{2} \): \( 1 \times 2 = 2 \) (not a multiple of 4)
For \( \frac{1}{5} \): \( 1 \times 5 = 5 \) (not a multiple of 4)
For \( \frac{2}{4} \): \( 2 \times 4 = 8 \) (is a multiple of 4)
For \( \frac{1}{8} \): \( 1 \times 8 = 8 \) (is a multiple of 4)
For \( \frac{2}{7} \): \( 2 \times 7 = 14 \) (not a multiple of 4)
For \( \frac{3}{6} \): \( 3 \times 6 = 18 \) (not a multiple of 4)
For \( \frac{4}{5} \): \( 4 \times 5 = 20 \) (is a multiple of 4)
For \( \frac{3}{9} \): \( 3 \times 9 = 27 \) (not a multiple of 4)
For \( \frac{4}{8} \): \( 4 \times 8 = 32 \) (is a multiple of 4)
For \( \frac{5}{7} \): \( 5 \times 7 = 35 \) (not a multiple of 4)
For \( \frac{6}{9} \): \( 6 \times 9 = 54 \) (not a multiple of 4)
For \( \frac{7}{8} \): \( 7 \times 8 = 56 \) (is a multiple of 4)
So, the possible fractions are \( \frac{1}{8}, \frac{2}{4}, \frac{4}{5}, \frac{4}{8}, \frac{7}{8} \). These fractions fit all the rules given. This type of problem helps improve logical thinking and number sense.
In simple words: We list all fractions where both numbers are single digits and the top number is smaller than the bottom. Then we check which of these fractions have numbers that add up to a multiple of 3. Finally, from that list, we find which ones have numbers that multiply to a multiple of 4.

🎯 Exam Tip: Break down clue-based problems step-by-step. Start by listing all possibilities from the first clue, then filter that list using the second clue, and so on.

 

Question 7. Add the difference between \( 1\frac{1}{3} \) and \( 3\frac{1}{6} \) and the difference between \( 4\frac{1}{6} \) and \( 2\frac{1}{3} \).
Answer: We need to calculate two differences and then add them together.

First, calculate the difference between \( 3\frac{1}{6} \) and \( 1\frac{1}{3} \):
Difference 1 \( = 3\frac{1}{6} - 1\frac{1}{3} \)
Convert to improper fractions:
\( = \frac{19}{6} - \frac{4}{3} \)
Find a common denominator, which is 6:
\( = \frac{19}{6} - \frac{4 \times 2}{3 \times 2} \)
\( = \frac{19}{6} - \frac{8}{6} \)
\( = \frac{19-8}{6} \)
\( = \frac{11}{6} \)

Second, calculate the difference between \( 4\frac{1}{6} \) and \( 2\frac{1}{3} \):
Difference 2 \( = 4\frac{1}{6} - 2\frac{1}{3} \)
Convert to improper fractions:
\( = \frac{25}{6} - \frac{7}{3} \)
Find a common denominator, which is 6:
\( = \frac{25}{6} - \frac{7 \times 2}{3 \times 2} \)
\( = \frac{25}{6} - \frac{14}{6} \)
\( = \frac{25-14}{6} \)
\( = \frac{11}{6} \)

Now, add the two differences:
Adding Difference \( = \frac{11}{6} + \frac{11}{6} \)
\( = \frac{11+11}{6} \)
\( = \frac{22}{6} \)
Simplify the fraction by dividing both numerator and denominator by 2:
\( = \frac{11}{3} \)
Convert to a mixed fraction:
\( = 3\frac{2}{3} \)
The final sum is \( 3\frac{2}{3} \). This problem shows how we can combine multiple fractional operations in a sequence.
In simple words: First, find the difference between the first two numbers. Then, find the difference between the next two numbers. Finally, add these two differences together.

🎯 Exam Tip: Always perform operations within parentheses or in the specified order. Remember to simplify fractions to their lowest terms at the end.

 

Question 8. What fraction is to be subtracted from \( 9\frac{3}{7} \) to get \( 3\frac{1}{5} \)?
Answer: Let the fraction to be subtracted be \( x \).
According to the problem, we can write the equation:
\( 9\frac{3}{7} - x = 3\frac{1}{5} \)
To find \( x \), we rearrange the equation:
\( x = 9\frac{3}{7} - 3\frac{1}{5} \)
Convert the mixed fractions to improper fractions:
\( x = \frac{66}{7} - \frac{16}{5} \)
Find a common denominator, which is 35 (the product of 7 and 5):
\( x = \frac{(66 \times 5) - (16 \times 7)}{7 \times 5} \)
\( x = \frac{330 - 112}{35} \)
\( x = \frac{218}{35} \)
Convert the improper fraction to a mixed fraction:
\( x = 6\frac{8}{35} \)
So, the fraction to be subtracted is \( 6\frac{8}{35} \). This type of problem is similar to finding a missing number in an equation and helps us practice working backwards.
In simple words: We want to find a fraction that, when taken away from \( 9\frac{3}{7} \), leaves \( 3\frac{1}{5} \). To do this, we subtract \( 3\frac{1}{5} \) from \( 9\frac{3}{7} \).

🎯 Exam Tip: When solving for an unknown fraction in a subtraction problem, always remember to isolate the unknown by moving other terms to the opposite side of the equation with inverse operations.

 

Question 9. The sum of two fractions is \( 5\frac{3}{9} \). If one of the fractions is \( 2\frac{3}{4} \), find the other fraction.
Answer: Let the other fraction be \( x \).
According to the problem, the sum of two fractions is \( 5\frac{3}{9} \). One fraction is \( 2\frac{3}{4} \). So, we can write the equation:
\( 2\frac{3}{4} + x = 5\frac{3}{9} \)
First, simplify \( 5\frac{3}{9} \) to \( 5\frac{1}{3} \).
\( 2\frac{3}{4} + x = 5\frac{1}{3} \)
To find \( x \), we subtract \( 2\frac{3}{4} \) from \( 5\frac{1}{3} \):
\( x = 5\frac{1}{3} - 2\frac{3}{4} \)
Convert the mixed fractions to improper fractions:
\( x = \frac{16}{3} - \frac{11}{4} \)
Find a common denominator, which is 12 (the product of 3 and 4):
\( x = \frac{(16 \times 4) - (11 \times 3)}{12} \)
\( x = \frac{64 - 33}{12} \)
\( x = \frac{31}{12} \)
Convert the improper fraction to a mixed fraction:
\( x = 2\frac{7}{12} \)
So, the other fraction is \( 2\frac{7}{12} \). This problem reinforces our ability to work with fractions in addition and subtraction contexts.
In simple words: We know the total of two fractions and one of the fractions. To find the other fraction, we subtract the known fraction from the total sum.

🎯 Exam Tip: Always simplify fractions (like \( 5\frac{3}{9} \) to \( 5\frac{1}{3} \)) before performing operations to make calculations easier and reduce errors.

 

Question 10. By what number should \( 3\frac{1}{16} \) be multiplied to get \( 9\frac{3}{16} \)?
Answer: Let the number be \( x \).
According to the problem, when \( 3\frac{1}{16} \) is multiplied by \( x \), the result is \( 9\frac{3}{16} \). We can write this as an equation:
\( 3\frac{1}{16} \times x = 9\frac{3}{16} \)
To find \( x \), we need to divide \( 9\frac{3}{16} \) by \( 3\frac{1}{16} \).
First, convert the mixed fractions to improper fractions:
\( 3\frac{1}{16} = \frac{(3 \times 16) + 1}{16} = \frac{48+1}{16} = \frac{49}{16} \)
\( 9\frac{3}{16} = \frac{(9 \times 16) + 3}{16} = \frac{144+3}{16} = \frac{147}{16} \)
Now, the equation becomes:
\( \frac{49}{16} \times x = \frac{147}{16} \)
To solve for \( x \), divide both sides by \( \frac{49}{16} \):
\( x = \frac{147}{16} \div \frac{49}{16} \)
To divide by a fraction, multiply by its reciprocal:
\( x = \frac{147}{16} \times \frac{16}{49} \)
The 16s cancel out:
\( x = \frac{147}{49} \)
Divide 147 by 49:
\( x = 3 \)
The number is 3. This problem shows how to reverse a multiplication to find a missing factor.
In simple words: To find the unknown number, divide the target number (\( 9\frac{3}{16} \)) by the given number (\( 3\frac{1}{16} \)).

🎯 Exam Tip: Always convert mixed fractions to improper fractions before performing multiplication or division to ensure correct calculations.

 

Question 11. Complete the fifth row in the Leibnitz triangle which is based on subtraction.
Answer: The Leibnitz triangle (also known as the Leibniz harmonic triangle) is constructed such that each element is the sum of the two elements directly below it. So, to find the elements of a row from the row below, we use subtraction: \( T_{n+1,k} = T_{n,k} - T_{n+1,k+1} \) (where \( T_{n,k} \) is an element in row \( n \) and column \( k \)). The first and last elements of each row \( n \) are \( \frac{1}{n} \).
The given triangle is:
\( \frac{1}{1} \)
\( \frac{1}{2} \ \ \frac{1}{2} \)
\( \frac{1}{3} \ \ \frac{1}{6} \ \ \frac{1}{3} \)
\( \frac{1}{4} \ \ \frac{1}{12} \ \ \frac{1}{12} \ \ \frac{1}{4} \)

We need to find the fifth row. For the fifth row (n=5):
The first element is \( \frac{1}{5} \).
The last element is \( \frac{1}{5} \).

Let's calculate the elements using the subtraction rule based on the 4th row elements:
\( T_{5,1} = \frac{1}{5} \)
\( T_{5,2} = T_{4,1} - T_{5,1} = \frac{1}{4} - \frac{1}{5} = \frac{5-4}{20} = \frac{1}{20} \)
\( T_{5,3} = T_{4,2} - T_{5,2} = \frac{1}{12} - \frac{1}{20} = \frac{5-3}{60} = \frac{2}{60} = \frac{1}{30} \)
\( T_{5,4} = T_{4,3} - T_{5,3} = \frac{1}{12} - \frac{1}{30} = \frac{5-2}{60} = \frac{3}{60} = \frac{1}{20} \)
\( T_{5,5} = T_{4,4} - T_{5,4} = \frac{1}{4} - \frac{1}{20} = \frac{5-1}{20} = \frac{4}{20} = \frac{1}{5} \)

So, the complete fifth row is \( \frac{1}{5}, \frac{1}{20}, \frac{1}{30}, \frac{1}{20}, \frac{1}{5} \). The pattern here is quite symmetric, which is a key feature of such mathematical triangles.

This is how the completed triangle looks with the fifth row:

In simple words: In this special triangle, each number is found by subtracting the number to its right below it from the number directly above it. We used this rule to find all the numbers in the fifth row.

🎯 Exam Tip: For problems involving number patterns or sequences, identify the rule of formation first. For Leibniz triangle, elements can be found using addition of the two elements below, or subtraction from the element above, working from right to left.

 

Question 12. A painter painted \( \frac{3}{8} \) of the wall of which one third is painted in yellow colour. What fraction is the yellow colour of the entire wall?
Answer: The painter painted \( \frac{3}{8} \) of the wall. Out of this painted area, \( \frac{1}{3} \) is yellow. To find the fraction of the entire wall that is yellow, we multiply these two fractions.
Fraction of wall painted \( = \frac{3}{8} \)
Fraction of painted wall that is yellow \( = \frac{1}{3} \)
Fraction of yellow colour of the entire wall \( = \frac{3}{8} \times \frac{1}{3} \)
Multiply the numerators and the denominators:
\( = \frac{3 \times 1}{8 \times 3} \)
Cancel out the common factor of 3:
\( = \frac{1}{8} \)
So, \( \frac{1}{8} \) of the entire wall is painted in yellow colour. This problem shows how fractions are used to describe parts of parts, which is a common application.
In simple words: To find out what part of the whole wall is yellow, we multiply the fraction of the wall that was painted by the fraction of that painted part that is yellow.

🎯 Exam Tip: When you need to find a "fraction of a fraction", it means you should multiply the two fractions together. Simplify before multiplying if possible to make calculations easier.

 

Question 13. A rabbit has to cover \( 26\frac{1}{4} \) m to fetch its food. If it covers \( 1\frac{3}{4} \) m in one jump, then how many jumps will it take to fetch its food?
Answer: We need to find out how many jumps the rabbit needs. The total distance to the food is \( 26\frac{1}{4} \) m, and the rabbit covers \( 1\frac{3}{4} \) m in each jump. To find the number of jumps, we divide the total distance by the distance covered in one jump.
Total distance \( = 26\frac{1}{4} \) m
Distance covered in one jump \( = 1\frac{3}{4} \) m
Number of jumps required \( = \frac{\text{Total distance}}{\text{Distance per jump}} \)
\( = 26\frac{1}{4} \div 1\frac{3}{4} \)
Convert the mixed fractions to improper fractions:
\( 26\frac{1}{4} = \frac{(26 \times 4) + 1}{4} = \frac{104+1}{4} = \frac{105}{4} \)
\( 1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{4+3}{4} = \frac{7}{4} \)
Now, perform the division:
\( = \frac{105}{4} \div \frac{7}{4} \)
To divide by a fraction, multiply by its reciprocal:
\( = \frac{105}{4} \times \frac{4}{7} \)
The 4s cancel out:
\( = \frac{105}{7} \)
Divide 105 by 7:
\( = 15 \)
The rabbit will take 15 jumps to fetch its food. This problem shows how division is used to count how many times one quantity fits into another.
In simple words: Divide the total distance the rabbit needs to go by the distance it covers in one jump. This will tell us the number of jumps needed.

🎯 Exam Tip: When dividing by a fraction, remember to multiply by its reciprocal. Always convert mixed numbers to improper fractions first for accurate division.

 

Question 14. Look at the picture and answer the following questions.
(i) What is the distance from the school to Library via Bus stop?
(ii) What is the distance between School and Library via Hospital?
(iii) Which is the shortest distance between (i) and (ii)?
(iv) The distance between School and Hospital is (blank) times the distance between School and Bus stop.
Answer: Based on the map shown:
- Distance from School to Bus stop = \( 3\frac{1}{4} \) km
- Distance from Bus stop to Library = \( 3\frac{1}{2} \) km
- Distance from Hospital to Library = \( 1\frac{1}{4} \) km
- Distance from School to Hospital = \( 4\frac{1}{2} \) km

(i) Distance from School to Library via Bus stop:
This is the sum of School to Bus stop and Bus stop to Library.
Distance \( = 3\frac{1}{4} + 3\frac{1}{2} \) km
Convert to improper fractions:
\( = \frac{13}{4} + \frac{7}{2} \) km
Find a common denominator, which is 4:
\( = \frac{13}{4} + \frac{7 \times 2}{2 \times 2} \) km
\( = \frac{13}{4} + \frac{14}{4} \) km
\( = \frac{13+14}{4} \) km
\( = \frac{27}{4} \) km
Convert to mixed fraction:
\( = 6\frac{3}{4} \) km

(ii) Distance between School and Library via Hospital:
This is the sum of School to Hospital and Hospital to Library.
Distance \( = 4\frac{1}{2} + 1\frac{1}{4} \) km
Convert to improper fractions:
\( = \frac{9}{2} + \frac{5}{4} \) km
Find a common denominator, which is 4:
\( = \frac{9 \times 2}{2 \times 2} + \frac{5}{4} \) km
\( = \frac{18}{4} + \frac{5}{4} \) km
\( = \frac{18+5}{4} \) km
\( = \frac{23}{4} \) km
Convert to mixed fraction:
\( = 5\frac{3}{4} \) km

(iii) Which is the shortest distance between (i) and (ii)?
Distance (i) = \( 6\frac{3}{4} \) km
Distance (ii) = \( 5\frac{3}{4} \) km
Since \( 5\frac{3}{4} < 6\frac{3}{4} \), the distance via Hospital is shorter.
Shortest distance is \( 5\frac{3}{4} \) km via Hospital.

(iv) The distance between School and Hospital is (blank) times the distance between School and Bus stop.
Distance from School to Hospital \( = 4\frac{1}{2} \) km \( = \frac{9}{2} \) km
Distance from School to Bus stop \( = 3\frac{1}{4} \) km \( = \frac{13}{4} \) km
To find how many times, we divide: \( \frac{\text{School to Hospital}}{\text{School to Bus stop}} \)
\( = \frac{9}{2} \div \frac{13}{4} \)
\( = \frac{9}{2} \times \frac{4}{13} \)
\( = \frac{9 \times 2}{13} \) (since \( \frac{4}{2} = 2 \))
\( = \frac{18}{13} \)
\( = 1\frac{5}{13} \)
So, the distance between School and Hospital is \( 1\frac{5}{13} \) times the distance between School and Bus stop. This exercise helps us to plan routes and compare distances in real-world situations.
In simple words: We calculated two different routes to the Library and found the shorter one. Then, we compared two specific distances to see how many times one distance was bigger than the other.

🎯 Exam Tip: For multi-part questions, calculate each part carefully step-by-step. For comparisons, make sure all numbers are in a comparable format (e.g., mixed fractions or improper fractions with common denominators).

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