Context: Manjunath usually mixes of coffee decoction with of milk to make filter coffee (ratio ). For 'stronger' coffee, he mixes of decoction with of milk (ratio ). Q. Why is this coffee stronger?

Understand the Question The strength of coffee depends on the proportion of coffee decoction in the total mixture (part-to-whole ratio). In both recipes, the total volume of coffee is . To find out why the second coffee is stronger, we calculate and compare the fraction (or percentage) of decoction present in the total volume for both recipes. Step 1 · Find the Proportion of Decoction in Normal Coffee Calculate the total volume: Calculate the proportion of decoction: Step 2 · Find the Proportion of Decoction in Stronger Coffee Calculate the total volume: Calculate the proportion of decoction: Step 3 · Compare the Proportions Comparing the two proportions: Since the second coffee mixture contains a greater fraction of decoction for the same total volume of , it tastes stronger. Answer > The second coffee is stronger because it contains a higher proportion of coffee decoction ( or ) compared to the normal coffee ( or ). Common Mistakes Confusing Part-to-Part with Part-to-Whole: Calculating and directly (decoction to milk) instead of finding the proportion of decoction in the entire mixture (). Ignoring Total Volume: Comparing only the amount of decoction ( vs ) without ensuring the total volumes are equal.

Proportional Reasoning - 1 | IT

Question 9

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?

Solution

Understand the Question

The strength of coffee depends on the proportion of coffee decoction in the total mixture (part-to-whole ratio).
In both recipes, the total volume of coffee is $50\text{ mL}$ .
To find out why the second coffee is stronger, we calculate and compare the fraction (or percentage) of decoction present in the total volume for both recipes.

Step 1 · Find the Proportion of Decoction in Normal Coffee

Calculate the total volume:

\begin{aligned} \text{Total volume} &= 15\text{ mL} + 35\text{ mL} \\[0.6em] &= 50\text{ mL} \end{aligned}

Calculate the proportion of decoction:

\begin{aligned} \text{Proportion of decoction} &= \dfrac{\text{Volume of decoction}}{\text{Total volume}} \\[0.6em] &= \dfrac{15\text{ mL}}{50\text{ mL}} \\[0.6em] &= \dfrac{3}{10} = 0.3 \text{ (or } 30\%\text{)} \end{aligned}

Step 2 · Find the Proportion of Decoction in Stronger Coffee

Calculate the total volume:

\begin{aligned} \text{Total volume} &= 20\text{ mL} + 30\text{ mL} \\[0.6em] &= 50\text{ mL} \end{aligned}

Calculate the proportion of decoction:

\begin{aligned} \text{Proportion of decoction} &= \dfrac{\text{Volume of decoction}}{\text{Total volume}} \\[0.6em] &= \dfrac{20\text{ mL}}{50\text{ mL}} \\[0.6em] &= \dfrac{2}{5} = 0.4 \text{ (or } 40\%\text{)} \end{aligned}

Step 3 · Compare the Proportions

Comparing the two proportions:

0.4 > 0.3 \quad (40\% > 30\%)

Since the second coffee mixture contains a greater fraction of decoction for the same total volume of $50\text{ mL}$ , it tastes stronger.

Answer

The second coffee is stronger because it contains a higher proportion of coffee decoction ( $0.4$ or $40\%$ ) compared to the normal coffee ( $0.3$ or $30\%$ ).

Common Mistakes

Confusing Part-to-Part with Part-to-Whole: Calculating $\dfrac{15}{35}$ and $\dfrac{20}{30}$ directly (decoction to milk) instead of finding the proportion of decoction in the entire mixture ( $\dfrac{\text{decoction}}{\text{total volume}}$ ).
Ignoring Total Volume: Comparing only the amount of decoction ( $20\text{ mL}$ vs $15\text{ mL}$ ) without ensuring the total volumes are equal.

Question 9

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