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

Understand the Question The strength or darkness of coffee depends on the concentration (proportion) of coffee decoction relative to the total volume of the mixture. To determine why the coffee is 'lighter', we calculate and compare the fraction of coffee decoction in the total mixture for both cases. Step 1 · Calculate Coffee Proportion for Normal Coffee Find the total volume of the normal coffee mixture: Calculate the proportion of coffee decoction: Step 2 · Calculate Coffee Proportion for Lighter Coffee Find the total volume of the lighter coffee mixture: Calculate the proportion of coffee decoction: Step 3 · Compare the Proportions Comparing the two proportions: Since the proportion of coffee decoction is lower in the second mixture ( compared to ), it has a lower concentration of coffee, making it lighter. Answer > The coffee is lighter because it has a lower proportion of coffee decoction ( or ) in the total mixture compared to normal coffee ( or ). Common Mistakes Part-to-Part vs. Part-to-Whole: Confusing the ratio of ingredients () with the proportion of coffee in the entire drink (). Concentration is always relative to the total volume. Ignoring Total Volume: Comparing only the amount of coffee decoction ( vs ) without verifying the total volume of the mixture.

Proportional Reasoning - 1 | IT

Question 10

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?

Solution

Understand the Question

The strength or darkness of coffee depends on the concentration (proportion) of coffee decoction relative to the total volume of the mixture.
To determine why the coffee is 'lighter', we calculate and compare the fraction of coffee decoction in the total mixture for both cases.

Step 1 · Calculate Coffee Proportion for Normal Coffee

Find the total volume of the normal coffee mixture:

\begin{aligned} \text{Total volume (normal)} &= \text{Coffee decoction} + \text{Milk} \\[0.6em] &= 15\text{ mL} + 35\text{ mL} \\[0.6em] &= 50\text{ mL} \end{aligned}

Calculate the proportion of coffee decoction:

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

Step 2 · Calculate Coffee Proportion for Lighter Coffee

Find the total volume of the lighter coffee mixture:

\begin{aligned} \text{Total volume (lighter)} &= \text{Coffee decoction} + \text{Milk} \\[0.6em] &= 10\text{ mL} + 40\text{ mL} \\[0.6em] &= 50\text{ mL} \end{aligned}

Calculate the proportion of coffee decoction:

\begin{aligned} \text{Proportion of coffee (lighter)} &= \dfrac{\text{Volume of coffee decoction}}{\text{Total volume}} \\[0.6em] &= \dfrac{10\text{ mL}}{50\text{ mL}} \\[0.6em] &= \dfrac{1}{5} = 0.2 \quad (20\%) \end{aligned}

Step 3 · Compare the Proportions

Comparing the two proportions: $0.2 < 0.3$

Since the proportion of coffee decoction is lower in the second mixture ( $20\%$ compared to $30\%$ ), it has a lower concentration of coffee, making it lighter.

Answer

The coffee is lighter because it has a lower proportion of coffee decoction ( $0.2$ or $20\%$ ) in the total mixture compared to normal coffee ( $0.3$ or $30\%$ ).

Common Mistakes

Part-to-Part vs. Part-to-Whole: Confusing the ratio of ingredients ( $15:35$ ) with the proportion of coffee in the entire drink ( $\dfrac{15}{15+35} = \dfrac{15}{50}$ ). Concentration is always relative to the total volume.
Ignoring Total Volume: Comparing only the amount of coffee decoction ( $10\text{ mL}$ vs $15\text{ mL}$ ) without verifying the total volume of the mixture.

Question 10

More questions in IT