1. Equivalent Ratios
Which of the following represents the same ratio as $4:7$?
A. $8:21$
B. $12:21$
C. $16:35$
D. $20:28$
Answer: B. $12:21$
2. Testing Proportionality
Two ratios can be considered proportional if:
A. Their corresponding terms have the same sum
B. Their cross products have equal values
C. Their numerators are always equal
D. Their denominators differ by the same amount
Answer: B. Their cross products have equal values
3. Find the Missing Term
If
\(\frac{x}{7}=\frac{4}{5},\)
then the value of $x$ is:
A. $\frac{20}{7}$
B. $\frac{28}{5}$
C. $\frac{28}{5}$
D. $\frac{35}{4}$
Answer: C. $\frac{28}{5}$
4. Cross Multiplication
For the proportion
\(\frac{3}{4}=\frac{6}{8},\)
which equation results from cross multiplication?
A. $3+8=4+6$
B. $3\times6=4\times8$
C. $3\times8=4\times6$
D. $3\times4=6\times8$
Answer: C. $3\times8=4\times6$
5. Simplifying a Ratio
Reduce $4:6:8$ to its simplest form.
A. $1:2:3$
B. $2:3:4$
C. $4:3:2$
D. $2:4:6$
Answer: B. $2:3:4$
6. HCF and Ratios
What is the HCF of $8$, $4$ and $21$?
A. 1
B. 2
C. 4
D. 8
Answer: A. 1
7. Representative Fraction
A map has an RF of $1:6,000,000$. What ground distance corresponds to 1 cm on the map?
A. 6 km
B. 60 km
C. 600 km
D. 6000 km
Answer: B. 60 km
8. Map Distance to Actual Distance
A map uses the scale $1:1,00,000$. If the distance between two places on the map is 5 cm, the actual distance is:
A. 500 m
B. 5 km
C. 50 km
D. 500 km
Answer: B. 5 km
9. Meaning of RF
The representative fraction of a map is used to compare:
A. Two different map areas
B. Map distance with actual distance
C. Actual area with map area
D. Map height with map width
Answer: B. Map distance with actual distance
10. Dividing Money in a Ratio
₹360 is to be distributed among three people in the ratio $2:4:6$. What is the share of the person with the largest ratio?
A. ₹60
B. ₹120
C. ₹150
D. ₹180
Answer: D. ₹180
11. Ratio Division
A total amount of ₹240 is divided in the ratio $2:3:5$. How much does the person with ratio 3 receive?
A. ₹48
B. ₹60
C. ₹72
D. ₹120
Answer: C. ₹72
12. Finding the Total Ratio
Three quantities are represented by the ratio $7:5:3$. What is the sum of the ratio terms?
A. 12
B. 15
C. 18
D. 21
Answer: B. 15
## Pie Charts and Angles
13. Complete Pie Chart
What is the total angle corresponding to an entire pie chart?
A. $90^\circ$
B. $180^\circ$
C. $270^\circ$
D. $360^\circ$
Answer: D. $360^\circ$
14. Fraction of a Pie Chart
A sector occupies $90^\circ$ of a pie chart. What fraction of the complete chart does it represent?
A. $\frac18$
B. $\frac14$
C. $\frac12$
D. $\frac34$
Answer: B. $\frac14$
you obtains this by calculating $\frac{90}{360}$.
15. Finding a Sector Angle
What angle should be assigned to a sector representing $\frac14$ of the observations in a pie chart?
A. $45^\circ$
B. $60^\circ$
C. $90^\circ$
D. $120^\circ$
Answer: C. $90^\circ$
16. Triangle Angle Ratio
The three angles of a triangle are in the ratio $2:3:4$. What is the largest angle?
A. $40^\circ$
B. $60^\circ$
C. $80^\circ$
D. $100^\circ$
Answer: C. $80^\circ$
17. Sum of Angles of a Triangle
If the angles of a triangle are expressed in a ratio, which total angle should be used to determine their actual values?
A. $90^\circ$
B. $180^\circ$
C. $270^\circ$
D. $360^\circ$
Answer: B. $180^\circ$
This uses the $180^\circ$ sum property of a triangle while finding an angle from a ratio.
## Inverse Proportion
18. Workers and Time
A fixed job takes 6 days when 10 workers are employed. If only 5 workers work at the same rate, how long will the job take?
A. 3 days
B. 6 days
C. 10 days
D. 12 days
Answer: D. 12 days
19. Identifying Inverse Proportion
Which situation best demonstrates inverse proportionality?
A. Increasing workers increases the time required
B. Increasing workers decreases the time required for the same work
C. Increasing the side of a square decreases its area
D. Increasing a quantity leaves its related quantity unchanged
Answer: B. Increasing workers decreases the time required for the same work
20. Doubling the Number of Workers
For a fixed amount of work, the number of workers is doubled. Assuming each worker works at the same rate, the completion time becomes:
A. Twice the original time
B. Four times the original time
C. Half the original time
D. Unchanged
Answer: C. Half the original time
21. Inverse Variation Formula
If $x$ varies inversely with $y$, which expression remains constant?
A. $\frac{x}{y}$
B. $x+y$
C. $x-y$
D. $xy$
Answer: D. $xy$
## Direct Proportion and Ratio
22. Direct Proportion
If $x$ is directly proportional to $y$, which quantity remains constant?
A. $x+y$
B. $x-y$
C. $\frac{x}{y}$
D. $xy$
Answer: C. $\frac{x}{y}$
23. Recognising Proportional Ratios
Which pair of ratios is proportional?
A. $2:3$ and $4:5$
B. $3:5$ and $9:15$
C. $4:7$ and $8:15$
D. $5:6$ and $10:15$
Answer: B. $3:5$ and $9:15$
24. Non-Proportional Ratios
Which pair does not represent equivalent ratios?
A. $2:3$ and $8:12$
B. $3:4$ and $9:12$
C. $4:5$ and $12:15$
D. $5:7$ and $15:22$
Answer: D. $5:7$ and $15:22$
25. Map Scale
A map scale is given as $1:50$. Which statement correctly interprets it?
A. 1 cm on the map represents 50 cm in reality
B. 50 cm on the map represents 1 cm in reality
C. 1 m on the map represents 50 km in reality
D. 50 km on the map represents 1 km in reality
Answer: A. 1 cm on the map represents 50 cm in reality
1. What does an RF of $1:5,00,000$ mean on a map?
Answer: It means that 1 cm on the map represents $5,00,000$ cm of actual distance on the ground.
2. Why do we divide a distance in centimetres by $1,00,000$ to convert it into kilometres?
Answer: Because $1$ km equals $1,00,000$ cm.
3. A map has an RF of $1:50,00,000$. What actual distance is represented by 4 cm?
Answer:
\(4\times50,00,000=2,00,00,000\text{ cm}\)
Since $1$ km $=1,00,000$ cm,
\(\frac{2,00,00,000}{1,00,000}=200\text{ km}.\)
Therefore, the actual distance is 200 km.
4. A mixture contains three ingredients in the ratio $6:3:1$. If the second ingredient is increased from 3 units to 9 units, what should happen to the other two quantities?
Answer: The ratio has been multiplied by 3. Therefore, the other quantities should also be multiplied by 3. The new quantities will be 18 units and 3 units.
5. Why must all ingredients be multiplied by the same factor when increasing a mixture?
Answer: Multiplying every term by the same factor preserves the original ratio and hence maintains the same composition of the mixture.
6. ₹900 is divided in the ratio $2:3:4$. Find the share corresponding to the ratio 4.
Answer:
Total parts:
\(2+3+4=9\)
Share:
\(\frac{4}{9}\times900=\boxed{₹400}\)
7. Why do we add all the terms of a ratio before dividing a total amount?
Answer: The sum gives the total number of equal ratio parts into which the given amount is divided.
8. Fifteen workers complete a task in 12 days. How many days will 20 workers need for the same task?
Answer: Workers and time are inversely proportional.
\(15\times12=20\times x\)
\(x=9\text{ days}\)
Therefore, 20 workers require 9 days.
9. What happens to the time required for a fixed task when the number of workers is increased?
Answer: The time required decreases, provided all workers work at the same rate. This is an example of inverse proportion.
10. Seven notebooks cost ₹140. Find the cost of 12 notebooks.
Answer:
Cost of one notebook:
\(\frac{140}{7}=₹20\)
Cost of 12 notebooks:
\(12\times20=\boxed{₹240}\)
## Conceptual Questions
11. Why are the number of workers and number of days considered inversely proportional for a fixed task?
Answer: If more workers are employed, the same amount of work can be completed in fewer days. Thus, one quantity increases while the other decreases.
12. If the number of workers is doubled, will the time always become half?
Answer: It will become half only when the total work remains fixed and each worker works at the same rate.
13. The angles of a triangle are in the ratio $2:3:4$. Explain how you would find the actual angles.
Answer: First add the ratio terms:
\(2+3+4=9\)
Since the sum of the angles of a triangle is $180^\circ$, one ratio part represents:
\(\frac{180^\circ}{9}=20^\circ\)
Therefore, the angles are:
\(2(20^\circ)=40^\circ\)
\(3(20^\circ)=60^\circ\)
\(4(20^\circ)=80^\circ\)
So the angles are $40^\circ,60^\circ$ and $80^\circ$.
14. Why is $180^\circ$ used when finding angles from a ratio in a triangle?
Answer: Because the sum of the three interior angles of every triangle is $180^\circ$.
15. What fraction of a pie chart is represented by a $90^\circ$ sector?
Answer:
\(\frac{90^\circ}{360^\circ}=\frac14\)
Therefore, the sector represents $\frac14$ of the total data.
16. Why is $360^\circ$ used when constructing a pie chart?
Answer: A pie chart represents a complete circle, and a complete circle has an angle of $360^\circ$.
17. In a class of 50 students, 10 students receive Grade A. What central angle should represent Grade A in a pie chart?
Answer:
\(\frac{10}{50}\times360^\circ=72^\circ\)
Therefore, Grade A should be represented by a $72^\circ$ sector.
18. Why does increasing every term of a ratio by the same factor preserve the ratio?
Answer: If each term is multiplied by the same number $k$,
\(a:b=(ka):(kb)\)
because the common factor $k$ cancels when the ratio is simplified.
19. A student says that increasing the number of workers from 15 to 30 will also increase the number of days from 12 to 24. Is the reasoning correct? Explain.
Answer: No. For a fixed task, workers and days are inversely proportional. If the workers are doubled, the required time becomes half:
\(15\times12=30\times x\)
Thus,
\(x=6\text{ days}.\)
20. Why can the unitary method be used to find the cost of 12 notebooks when the cost of 7 notebooks is known?
Answer: Because the notebooks are assumed to have the same individual price. We can first determine the cost of one notebook and then multiply it by 12. This is a direct-proportion situation.
Higher-Order Conceptual Questions
21. A map shows two towns 8 cm apart. Another map shows the same towns 4 cm apart. Can we conclude that both maps have the same scale? Explain.
Answer: No. The map distances can be different because the maps may have different scales. We need the RF or scale of each map to compare them.
22. A student calculates the angle corresponding to 15 students in a class of 50 as $\frac{15}{50}\times180^\circ$. Identify the mistake.
Answer: The student has used $180^\circ$, which is the total angle of a triangle. For a pie chart, the complete angle is $360^\circ$. The correct calculation is:
\(\frac{15}{50}\times360^\circ=108^\circ.\)
23. Why would multiplying only one ingredient in a mixture change the nature of the mixture?
Answer: Multiplying only one ingredient changes the original ratio between the ingredients. Therefore, the composition and hence the nature of the mixture will change.
24. If 18 workers complete a task in 10 days, why would fewer than 18 workers require more than 10 days for the same task?
Answer: The amount of work is fixed. Reducing the number of workers reduces the work completed per day, so more days are required. This shows inverse proportionality.
25. A pie-chart sector has an angle of $180^\circ$. What does this tell us about the corresponding data?
Answer: Since
\(\frac{180^\circ}{360^\circ}=\frac12,\)
the sector represents half of the total data, or 50%.