Proportional Reasoning - 1 | IT

Question 7

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

Solution

Understand the Question

Two shapes are proportional (similar) if the ratio of their corresponding dimensions remains the same.
Given that the blackboard has a width-to-height ratio of $2:1$ , any rectangle whose width is twice its height ( $W = 2H$ ) will be proportional to the blackboard and fit in a notebook.

Step 1 · Understand the Blackboard's Ratio

Let $W_B$ be the blackboard's width and $H_B$ be its height. Diagram 1

Given the ratio of width to height is $2:1$ $\dfrac{W_B}{H_B} = \dfrac{2}{1} \implies W_B = 2 \times H_B$

Step 2 · Choose Dimensions for the Notebook Rectangle

Let $W_N$ and $H_N$ be the width and height of the notebook rectangle.

For the rectangle to be proportional $\dfrac{W_N}{H_N} = \dfrac{2}{1} \implies W_N = 2 \times H_N$

Choosing a suitable height, $H_N = 5\text{ cm}$

\begin{aligned} W_N &= 2 \times 5\text{ cm} \\[0.6em] &= 10\text{ cm} \end{aligned}

Verifying the ratio

\begin{aligned} \dfrac{\text{Width}}{\text{Height}} &= \dfrac{10\text{ cm}}{5\text{ cm}} \\[0.6em] &= \dfrac{2}{1} \end{aligned}

Answer

Yes, a rectangle can be drawn with $\text{Width} = 10\text{ cm}$ and $\text{Height} = 5\text{ cm}$ .

Common Mistakes

Inverting the Ratio: Confusing width-to-height with height-to-width, leading to a ratio of $1:2$ instead of $2:1$ .
Non-Proportional Scaling: Adding equal lengths to both dimensions instead of multiplying both by the same scaling factor.

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?