Proportional Reasoning - 1 | A

Question 2

Go to the market and collect the prices of different sizes of shampoo containers of the same shampoo and create a table like the one given below. See if the volume of shampoo is proportional to the price.

Solution

Understand the Question

Two quantities are proportional if the ratio of one quantity to the other remains constant throughout (i.e. $\dfrac{\text{Price}}{\text{Volume}} = \text{constant}$ ).
To determine if the volume of shampoo is proportional to its price, we calculate the unit rate (price per $\text{mL}$ ) for each container size: $\text{Price per mL} = \dfrac{\text{Price}}{\text{Volume}}$
If the unit rates are equal across all sizes, the quantities are directly proportional; if they differ, they are not proportional.

Step 1 · Calculate Price per mL for Sachet

Diagram 1

\begin{aligned} \text{Price per mL (Sachet)} &= \dfrac{\text{Price}}{\text{Volume}} \\[0.6em] &= \dfrac{\text{₹}2}{6\text{ mL}} \\[0.6em] &= 0.3333\dots\text{ ₹/mL} \\[0.6em] &\approx 0.33\text{ ₹/mL} \end{aligned}

Step 2 · Calculate Price per mL for Small Bottle

Diagram 2

\begin{aligned} \text{Price per mL (Small Bottle)} &= \dfrac{\text{Price}}{\text{Volume}} \\[0.6em] &= \dfrac{\text{₹}154}{180\text{ mL}} \\[0.6em] &= 0.8555\dots\text{ ₹/mL} \\[0.6em] &\approx 0.86\text{ ₹/mL} \end{aligned}

Step 3 · Calculate Price per mL for Medium Bottle

Diagram 3

\begin{aligned} \text{Price per mL (Medium Bottle)} &= \dfrac{\text{Price}}{\text{Volume}} \\[0.6em] &= \dfrac{\text{₹}276}{340\text{ mL}} \\[0.6em] &= 0.8117\dots\text{ ₹/mL} \\[0.6em] &\approx 0.81\text{ ₹/mL} \end{aligned}

Step 4 · Calculate Price per mL for Large Bottle

Diagram 4

\begin{aligned} \text{Price per mL (Large Bottle)} &= \dfrac{\text{Price}}{\text{Volume}} \\[0.6em] &= \dfrac{\text{₹}540}{1000\text{ mL}} \\[0.6em] &= 0.54\text{ ₹/mL} \end{aligned}

Step 5 · Compare Price per mL Values

Summarizing the unit prices:

Sachet: $0.33\text{ ₹/mL}$
Small Bottle: $0.86\text{ ₹/mL}$
Medium Bottle: $0.81\text{ ₹/mL}$
Large Bottle: $0.54\text{ ₹/mL}$

Since the ratio $\dfrac{\text{Price}}{\text{Volume}}$ is not constant across all sizes, the volume of shampoo is not proportional to its price.

Answer

No, the volume of shampoo is not proportional to the price because the price per $\text{mL}$ varies across different container sizes.

Common Mistakes

Assuming Larger Always Means Cheaper Unit Price: Students often assume unit rates strictly decrease as bottle size increases, but packaging strategies (like low-cost promotional sachets) may result in unexpected variations.
Inverting the Ratio: Dividing $\dfrac{\text{Volume}}{\text{Price}}$ ( $\text{mL/₹}$ ) for some items and $\dfrac{\text{Price}}{\text{Volume}}$ ( $\text{₹/mL}$ ) for others. Keep the unit rate definition consistent across all comparisons.

Question 2

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