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

Understand the Question Two quantities are directly proportional if their ratio remains constant throughout. Here, for price and volume to be proportional, the unit price (price per mL) must be equal for both the sachet and the small bottle. Alternatively, the ratio of volumes must be equal to the ratio of prices . We can evaluate whether they are proportional by comparing their unit prices. Step 1 · Calculate Unit Price for the Sachet For the sachet, volume and price . Step 2 · Calculate Unit Price for the Small Bottle For the small bottle, volume and price . Step 3 · Compare the Unit Prices Comparing the two unit prices: Unit price for sachet Unit price for small bottle Since the price per unit volume is not equal, the ratio of the prices is not proportional to the ratio of the volumes. Answer > The ratio of prices is not proportional to the ratio of volumes because the unit price (price per mL) is different for the sachet and the bottle . Common Mistakes Assuming Proportionality from Simultaneous Increase: Believing that because both volume and price increase, they must be proportional. Proportionality requires the constant ratio to be strictly identical. Ratio Comparison Error: Not simplifying the fractions and , which directly proves .

Proportional Reasoning - 1 | IT

Question 14

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?

Solution

Understand the Question

Two quantities are directly proportional if their ratio remains constant throughout. Here, for price and volume to be proportional, the unit price (price per mL) must be equal for both the sachet and the small bottle.
Alternatively, the ratio of volumes $\left(\dfrac{6}{180} = \dfrac{1}{30}\right)$ must be equal to the ratio of prices $\left(\dfrac{2}{154} = \dfrac{1}{77}\right)$ .
We can evaluate whether they are proportional by comparing their unit prices.

Step 1 · Calculate Unit Price for the Sachet

For the sachet, volume $= 6\text{ mL}$ and price $= \text{₹}2$ . Diagram 1

\begin{aligned} \text{Unit price for sachet} &= \dfrac{\text{Price}}{\text{Volume}} \\[0.6em] &= \dfrac{\text{₹}2}{6\text{ mL}} \\[0.6em] &= \dfrac{\text{₹}1}{3\text{ mL}} \\[0.6em] &\approx \text{₹}0.333\text{ per mL} \end{aligned}

Step 2 · Calculate Unit Price for the Small Bottle

For the small bottle, volume $= 180\text{ mL}$ and price $= \text{₹}154$ .

\begin{aligned} \text{Unit price for small bottle} &= \dfrac{\text{Price}}{\text{Volume}} \\[0.6em] &= \dfrac{\text{₹}154}{180\text{ mL}} \\[0.6em] &= \dfrac{\text{₹}77}{90\text{ mL}} \\[0.6em] &\approx \text{₹}0.856\text{ per mL} \end{aligned}

Step 3 · Compare the Unit Prices

Comparing the two unit prices:

Unit price for sachet $\approx \text{₹}0.333\text{ per mL}$
Unit price for small bottle $\approx \text{₹}0.856\text{ per mL}$

$\dfrac{2}{6} \neq \dfrac{154}{180} \implies \dfrac{1}{3} \neq \dfrac{77}{90}$

Since the price per unit volume is not equal, the ratio of the prices is not proportional to the ratio of the volumes.

Answer

The ratio of prices is not proportional to the ratio of volumes because the unit price (price per mL) is different for the sachet $(\approx \text{₹}0.333/\text{mL})$ and the bottle $(\approx \text{₹}0.856/\text{mL})$ .

Common Mistakes

Assuming Proportionality from Simultaneous Increase: Believing that because both volume and price increase, they must be proportional. Proportionality requires the constant ratio $\dfrac{\text{Price}}{\text{Volume}}$ to be strictly identical.
Ratio Comparison Error: Not simplifying the fractions $\dfrac{6}{180} = \dfrac{1}{30}$ and $\dfrac{2}{154} = \dfrac{1}{77}$ , which directly proves $\dfrac{1}{30} \neq \dfrac{1}{77}$ .

Question 14

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