Proportional Reasoning - 1 | IT

Question 2

Can you check by what factors the width and height of image D change as compared to image A? Are the factors the same?

Solution

Understand the Question

To determine the factor by which dimensions change from Image A to Image D, divide each dimension (width and height) of Image D by the corresponding dimension of Image A:
- $\text{Factor} = \dfrac{\text{Dimension of Image D}}{\text{Dimension of Image A}}$
From the given data:
- Image A: $\text{Width} = 60\text{ mm}$ , $\text{Height} = 40\text{ mm}$
- Image D: $\text{Width} = 90\text{ mm}$ , $\text{Height} = 60\text{ mm}$

Step 1 · Calculate Width Factor

Diagram 1

Given $\text{Width of Image A} = 60\text{ mm}$ and $\text{Width of Image D} = 90\text{ mm}$

\begin{aligned} \text{Width factor} &= \dfrac{\text{Width of Image D}}{\text{Width of Image A}} \\[0.6em] &= \dfrac{90\text{ mm}}{60\text{ mm}} \\[0.6em] &= \dfrac{9}{6} \\[0.6em] &= \dfrac{3}{2} = 1.5 \end{aligned}

Step 2 · Calculate Height Factor

Diagram 2

Given $\text{Height of Image A} = 40\text{ mm}$ and $\text{Height of Image D} = 60\text{ mm}$

\begin{aligned} \text{Height factor} &= \dfrac{\text{Height of Image D}}{\text{Height of Image A}} \\[0.6em] &= \dfrac{60\text{ mm}}{40\text{ mm}} \\[0.6em] &= \dfrac{6}{4} \\[0.6em] &= \dfrac{3}{2} = 1.5 \end{aligned}

Step 3 · Compare the Factors

Comparing the two calculated scale factors: $\text{Width factor} = 1.5$ $\text{Height factor} = 1.5$

Since both factors equal $1.5$ (or $\dfrac{3}{2}$ ), the factors are the same.

Answer

The width and height of Image D both change by a factor of $1.5$ (or $\dfrac{3}{2}$ ) compared to Image A. Yes, the factors are the same.

Common Mistakes

Inverting the Ratio: Dividing the original dimension by the new dimension (e.g., $\dfrac{60}{90} = \dfrac{2}{3}$ ) instead of $\dfrac{\text{New}}{\text{Original}}$ ( $\dfrac{90}{60} = 1.5$ ).
Mixing Dimensions: Dividing the width of Image D by the height of Image A instead of comparing corresponding dimensions (width to width, height to height).

More questions in IT

Q. Which images look similar and which ones look different?

Can you check by what factors the width and height of image D change as compared to image A? Are the factors the same?

By what factor should we multiply the ratio $60 : 40$ (image A) to get $90 : 60$ (image D)?

In my school, there are 5 teachers and 170 students. The ratio of teachers to students in my school is $5 : 170$ . Count the number of teachers and students in your school. What is the ratio of teachers to students in your school? Write it below.

Is the teacher-to-student ratio in your school proportional to the one in my school?

Measure the width and height (to the nearest cm) of the blackboard in your classroom. What is the ratio of width to height of the blackboard?

Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard?

Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?

Context: Manjunath usually mixes $15\text{ mL}$ of coffee decoction with $35\text{ mL}$ of milk to make filter coffee (ratio $15 : 35$ ). For 'stronger' coffee, he mixes $20\text{ mL}$ of decoction with $30\text{ mL}$ of milk (ratio $20 : 30$ ).

Q. Why is this coffee stronger?

Q10

Context: Manjunath usually mixes $15\text{ mL}$ of coffee decoction with $35\text{ mL}$ of milk to make filter coffee (ratio $15 : 35$ ). For 'lighter' coffee, he mixes $10\text{ mL}$ of coffee and $40\text{ mL}$ of milk, making the ratio $10 : 40$ .

Q. Why is this coffee lighter?

Q11

The following table shows the different ratios in which Manjunath mixes coffee decoction with milk. Write in the last column if the coffee is stronger or lighter than the regular coffee.

Q12

Does the drawing look more realistic if the ratios are proportional? Why? Why not?

Q13

Puneeth’s father went from Lucknow to Kanpur in 2 hours by riding his motorcycle at a speed of 50 km/h. If he drives at 75 km/h, how long will it take him to reach Kanpur? Can we form this problem as a proportion—

Would it take Puneeth’s father more time or less time to reach Kanpur? Think about it.

Q14

Context: The volume of a sachet is 6 mL and its price is ₹2. The volume of a small bottle is 180 mL and its price is ₹154. The ratio of the volume of a sachet to a small bottle is $6 : 180$ . The ratio of their prices is $2 : 154$ .

Q. Why do you think that the ratio of the prices is not proportional to the ratio of the volumes?

Q15

Context: You have 12 countable objects or counters to share between the two of you.

Q. If you divide them equally, what is the ratio of the number of counters with each of you?

Q16

Context: You have 12 countable objects or counters to share between the two of you.

Q. If your partner gets 5 counters, how many objects will you get? What is the ratio of the counters?