By what factor should we multiply the ratio (image A) to get (image D)?

Understand the Question To find the factor by which the ratio (Image A) is multiplied to get (Image D), we compare the corresponding terms of both ratios. The scaling factor is obtained by dividing the dimensions of Image D by the corresponding dimensions of Image A. Step 1 · Calculate the Scaling Factor Given dimensions: Image A: , Image D: , Calculate the factor for width () Calculate the factor for height () Since , both dimensions scale by the same factor. Answer > (or ) Common Mistakes Inverting the Ratio: Dividing Image A's dimensions by Image D's dimensions gives instead of . Additive vs. Multiplicative Thinking: Adding to and to instead of finding the multiplicative scale factor required for ratios.

Proportional Reasoning - 1 | IT

Question 3

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

Solution

Understand the Question

To find the factor by which the ratio $60 : 40$ (Image A) is multiplied to get $90 : 60$ (Image D), we compare the corresponding terms of both ratios.
The scaling factor is obtained by dividing the dimensions of Image D by the corresponding dimensions of Image A.

Step 1 · Calculate the Scaling Factor

Given dimensions:

Image A: $\text{Width} = 60\text{ mm}$ , $\text{Height} = 40\text{ mm}$
Image D: $\text{Width} = 90\text{ mm}$ , $\text{Height} = 60\text{ mm}$

Calculate the factor for width ( $F_w$ )

\begin{aligned} F_w &= \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} \end{aligned}

Calculate the factor for height ( $F_h$ )

\begin{aligned} F_h &= \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} \end{aligned}

Since $F_w = F_h = \dfrac{3}{2} = 1.5$ , both dimensions scale by the same factor.

Answer

$\dfrac{3}{2}$ (or $1.5$ )

Common Mistakes

Inverting the Ratio: Dividing Image A's dimensions by Image D's dimensions gives $\dfrac{60}{90} = \dfrac{2}{3}$ instead of $\dfrac{90}{60} = \dfrac{3}{2}$ .
Additive vs. Multiplicative Thinking: Adding $30$ to $60$ and $20$ to $40$ instead of finding the multiplicative scale factor required for ratios.

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Q10

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Q11

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Q12

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Q13

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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 $2 : 154$ .

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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?