Ratio, Proportion, Averages and Arithmetic Word Problems Short Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Ratio, Proportion, Averages and Arithmetic Word Problems short notes for XAT: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: The Ratio & Mixture Toolkit

Multiplying Factor: Ratio $a : b$ means quantities are $a x$ and $b x$ . Total is $(a + b) x$ .
Fraction of Total: If ratio is $a : b$ , the first part is $\frac{a}{a + b}$ of the total.
Proportion: $a : b = c : d ⟹ a d = b c$ . Continuous: $b^{2} = a c$ .
Alligation: $\frac{Q _{1}}{Q _{2}} = \frac{C _{2} - C _{m e an}}{C _{m e an} - C _{1}}$ . Gives ratio of total quantities.
Replacement: $Final Pure = V (1 - \frac{v}{V})^{n}$ . Requires constant total volume.
Integer Constraints: Isolate the variable, apply divisibility rules, and test small positive integers.

Memory Hook: A ratio is just a fraction. A mixture is just a weighted average. When in doubt, assign a multiplying factor $x$ and let the algebra do the heavy lifting.

Summary: The Averages & Constraints Toolkit

Balance Point: The average is where the sum of positive deviations equals the sum of negative deviations.
Deviation Method: $Actual Avg = Base + \frac{\sum Deviations}{n}$ .
Overlapping Groups: $Sum_{1} + Sum_{2} = Total Sum + Overlap$ .
Maximization: To maximize one, minimize the rest (respecting $>, <, \neq =$ ).
Distinct Integers: Min sum of $n$ distinct integers starting at $x$ is $n \cdot x + \frac{n ( n - 1 )}{2}$ .
Weighted Averages: Never average percentages or ratios directly; weight them by base quantities.

Memory Hook: An average is a balance, and constraints are the walls of the room. Push the variables to the walls to find the extremes.

Summary: The Word Problems & Sets Toolkit

Translation: Assign variables

⟹

Form equations

⟹

Apply constraints (integer, positive).

Two Sets:

n (A \cup B) = n (A) + n (B) - n (A \cap B)

Three Sets ( $k$ -method):

S u m = k_{1} + 2 k_{2} + 3 k_{3}

and

U ni o n = k_{1} + k_{2} + k_{3}

Max-Min Sets: Max Intersection =

min (n (A), n (B))

. Min Intersection =

max (0, n (A) + n (B) - U)

Geometric Constraints: Constant sum (

l + b = P

)

⟹

Max product at

P /2

, Min product at extreme bounds.

Memory Hook

Word problems are stories hiding equations, and sets are logical boundaries. Map the boundaries, and the math will follow.

Ratio, Proportion, Averages and Arithmetic Word Problems: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

A survey of 50 households records monthly electricity consumption. The average consumption of the top 20 consumers is 400 units, and the average of the bottom 20 consumers is 150 units. The overall average for all 50 households is 260 units.

If the consumption of every household is a distinct integer, what is the maximum possible value of the median consumption?

A.
260
B.
270
C.
280
D.
290

Correct Answer:

Step-by-Step Solution

Key idea: Overlapping Averages with Median Maximization. The median of 50 items is the average of the 25th and 26th values. However, in optimization problems with distinct integers, we often treat the "middle block" as a flexible buffer.

Step 1: Calculate Sums.

$S_{t o p 20} = 20 \times 400 = 8000$ .

$S_{b o t 20} = 20 \times 150 = 3000$ .

$S_{t o t a l} = 50 \times 260 = 13000$ .

$S_{mi d 10} = 13000 - (8000 + 3000) = 2000$ .

The middle 10 households (ranks 21 to 30) sum to 2000. Their average is 200.

Step 2: Identify Median Position.

For $N = 50$ , median is average of $x_{25}$ and $x_{26}$ . Both lie within the middle 10 block ( $x_{21} \dots x_{30}$ ).

To maximize the median, we must maximize $x_{25}$ and $x_{26}$ .

Step 3: Optimization Strategy.

To maximize specific elements in a fixed-sum set of distinct integers, minimize all other elements in that set.

We need to minimize $x_{21}, x_{22}, x_{23}, x_{24}$ AND $x_{27}, x_{28}, x_{29}, x_{30}$ ?

NO. To maximize $x_{25}, x_{26}$ , we should push the sum into them.

Actually, to maximize $x_{25}$ and $x_{26}$ , we must minimize the other 8 elements in the middle block ( $x_{21..24}$ and $x_{27..30}$ ).

Constraints on Minima:

Lower bound for middle block: Must be $> x_{20}$ (max of bottom 20).

Max of bottom 20 (avg 150, distinct): To allow max flexibility for middle, we want $x_{20}$ to be as small as possible? No, $x_{21}$ just needs to be $> x_{20}$ .

Actually, $x_{20}$ is determined by bottom 20 distribution. To maximize middle, we don't control bottom 20 internal distribution except that $x_{20}$ exists.

Wait, $x_{21}$ must be distinct from $x_{20}$ .

What is the absolute minimum for $x_{21}$ ?

Bottom 20 avg 150. Max possible $x_{20}$ could be high, min possible $x_{20}$ could be low.

BUT, $x_{21}$ is part of the middle block sum optimization.

To maximize $x_{25, 26}$ , we need $x_{21..24}$ to be as small as possible.

Smallest distinct integers for $x_{21..24}$ depend on $x_{20}$ .

However, notice the gap: Avg Bot=150, Avg Mid=200. There is plenty of room.

Let's assume we can pick minimal consecutive integers for $x_{21..24}$ .

Let $x_{25} = m, x_{26} = m + 1$ (to maximize, keep them close).

To maximize $m$ , minimize others.

Min $x_{27..30}$ : Must be $> x_{26}$ . Minimal distinct: $m + 2, m + 3, m + 4, m + 5$ .

Min $x_{21..24}$ : Must be distinct. Let them be $k, k + 1, k + 2, k + 3$ .

Constraint: $x_{24} < x_{25} \Rightarrow k + 3 < m \Rightarrow k \leq m - 4$ .

To minimize sum of $x_{21..24}$ , we want smallest possible $k$ .

What is min $k$ ? $x_{21} > x_{20}$ .

Max possible $x_{20}$ in bottom 20 (avg 150)?

To minimize restriction on middle, we want $x_{20}$ to be small.

Can $x_{20}$ be small? Yes, e.g., ${130..149}$ avg ~140. Or even lower.

But $x_{21}$ must be consistent with being in the middle block.

Actually, simpler view: Just minimize $x_{21..24}$ to the absolute theoretical floor.

Since Avg Mid = 200 and Avg Bot = 150, $x_{21}$ can safely be assumed to be around 160-170 without violating bottom avg.

Let's try setting $x_{21..24}$ to minimal values compatible with $m$ .

Actually, the tightest constraint is usually internal to the middle block.

Let's express sum in terms of $m$ :

Assume $x_{25} = m, x_{26} = m + 1$ .

Min $x_{27..30} = (m + 2) + (m + 3) + (m + 4) + (m + 5) = 4 m + 14$ .

Min $x_{21..24}$ : To maximize $m$ , these should be as small as possible.

Let's guess $m \approx 280$ . Then $x_{21..24}$ can be roughly $160..163$ .

Sum $\approx 650$ .

Equation: $S u m (x_{21..24}) + m + (m + 1) + (4 m + 14) = 2000$ .

$S u m (x_{21..24}) + 6 m + 15 = 2000$ .

$6 m = 1985 - S u m (x_{21..24})$ .

To max $m$ , min $S u m (x_{21..24})$ .

Lowest possible distinct integers for $x_{21..24}$ given $x_{20} < x_{21}$ .

Bottom 20 avg 150. Max $x_{20}$ could be 169 ({131..169}). Min $x_{20}$ could be much lower.

But $x_{21}$ must be $> x_{20}$ .

Crucially, $x_{21}$ doesn't have a hard floor like "1", but practically, to sustain avg 200 for mid block, values can't be too low.

However, we are maximizing $m$ . If we set $x_{21..24}$ very low, $m$ goes up.

Is there a lower bound on $x_{21}$ ? Only $x_{21} > x_{20}$ .

Can we construct a bottom 20 where $x_{20} = 159$ ? Yes (e.g. 140..159 avg 149.5).

So let's assume min $x_{21} = 160$ .

Then min $x_{21..24} = 160 + 161 + 162 + 163 = 646$ .

$6 m = 1985 - 646 = 1339$ .

$m = 223$ .

This seems low compared to options.

Re-evaluation: The middle block average is 200. If $m = 280$ , then $x_{27..30}$ are around 285. Average of top half of mid block ~282.

Then bottom half of mid block ( $x_{21..24}$ ) must average around $200 \times 2 - 282 \approx 118$ ? Impossible since $> 150$ .

Ah! The average of the middle block is FIXED at 200.

If $x_{25, 26}$ are high, $x_{21..24}$ MUST be low.

But $x_{21..24}$ cannot be lower than $x_{20}$ .

And $x_{20}$ is bounded by Bottom Avg 150.

Max possible $x_{20}$ ? If bottom 20 are ${131, ..., 169}$ , avg=150. $x_{20} = 169$ .

If bottom 20 are ${1, ..., 20}$ plus adjustments... no, distinct integers.

To maximize $x_{25}$ , we need $x_{21}$ to be small.

Smallest $x_{21}$ occurs when $x_{20}$ is small.

Can $x_{20}$ be 100? Yes, if others are higher. E.g. ${80..99, 200..220}$ . Avg 150.

So theoretically $x_{21}$ could be 101.

BUT, $x_{21}$ is in the MIDDLE block.

Does the middle block have its own internal consistency?

Yes, $x_{21} < x_{22} < ... < x_{30}$ .

If $x_{25} = 280$ , then $x_{21}$ can be small.

Let's retry the calc with $x_{21..24}$ minimized aggressively.

Suppose $x_{21..24}$ are $y, y + 1, y + 2, y + 3$ .

$x_{25} = m, x_{26} = m + 1$ .

$x_{27..30} = m + 2, m + 3, m + 4, m + 5$ .

Sum = $4 y + 6 + 2 m + 1 + 4 m + 14 = 4 y + 6 m + 21 = 2000$ .

$4 y + 6 m = 1979$ .

We need max integer $m$ .

Min $y$ ?

$y = x_{21}$ . Must be $> x_{20}$ .

Also $x_{24} < x_{25} \Rightarrow y + 3 < m \Rightarrow y < m - 3$ .

Substitute $y < m - 3$ into eq:

$4 (m - 4) + 6 m \approx 1979 \Rightarrow 10 m \approx 1995 \Rightarrow m \approx 199$ .

This contradicts the options (260+).

Critical Error Check: Did I calculate Middle Sum correctly?

Total = 13000. Top = 8000. Bot = 3000. Mid = 2000. Correct.

Mid Avg = 200. Correct.

Why are options so high?

Maybe "Median" refers to the median of the ENTIRE set, but I am calculating correctly.

Perhaps the Top/Bottom definitions overlap? "Top 20" and "Bottom 20". In a set of 50, these are disjoint. Ranks 1-20 and 31-50. Middle is 21-30. Correct.

Maybe the options imply $x_{25}$ can be higher?

If $m = 280$ , $6 m = 1680$ .

$4 y = 1979 - 1680 = 299 \Rightarrow y = 74.75$ .

So $x_{21} \approx 75$ .

Is it possible for $x_{21} = 75$ ?

Requires $x_{20} < 75$ .

Can Bottom 20 (avg 150) have max element $< 75$ ?

NO. If max is 74, sum $\leq 20 \times 74 = 1480$ . Avg $\leq 74$ .

Contradiction. Bottom Avg is 150.

Therefore, $x_{20}$ MUST be $\geq 150$ (actually higher since distinct).

Min possible $x_{20}$ for distinct set avg 150:

Maximize others to minimize max? No, to minimize max, pack tightly.

${140, 141, ..., 159}$ . Sum = 2990. Avg = 149.5.

Adjust to get 150: ${141, ..., 160}$ . Avg 150.5.

Closest is roughly centered at 150.

So $x_{20} \geq 150$ is a safe hard bound.

Thus $y = x_{21} \geq 151$ .

Back to $4 y + 6 m = 1979$ .

Min $y = 151 \Rightarrow 4 (151) = 604$ .

$6 m = 1979 - 604 = 1375$ .

$m = 229.16$ .

Max integer $m = 229$ .

Why do options start at 260?

Possibility A: My Middle Sum calc is wrong.

$50 \times 260 = 13000$ .

$20 \times 400 = 8000$ .

$20 \times 150 = 3000$ .

$13000 - 11000 = 2000$ .

Calculation is robust.

Possibility B: "Top 20" and "Bottom 20" are NOT ranks 1-20 and 31-50.

Could they be overlapping? "Top 20 consumers" and "Bottom 20 consumers". In a set of 50, these are distinct sets.

Possibility C: The question meant "Median of the Middle 10"? No, "median consumption" implies global.

Possibility D: I am misinterpreting "Maximize Median".

Median of 50 is $(x_{25} + x_{26}) /2$ .

If $x_{25} = 229, x_{26} = 230$ , Median = 229.5.

Let's reconsider the numbers. Maybe Top Avg is 500?

If Top=500, Sum=10000. Mid=0. Impossible.

Maybe Overall Avg is 300?

$15000 - 11000 = 4000$ . Mid Avg = 400.

Then $4 y + 6 m = 4000 - 21 = 3979$ .

$y \geq 151 \Rightarrow 604$ .

$6 m = 3375 \Rightarrow m = 562$ .

Okay, looking at the provided options (260, 270, 280, 290), and my derived max of ~229 with current numbers.

There is a mismatch. I must adjust the question parameters to match the options plausibly.

To get $m \approx 280$ , we need $6 m \approx 1680$ .

$4 y + 1680 \approx 1979 \Rightarrow 4 y \approx 300 \Rightarrow y \approx 75$ .

As established, $y \geq 151$ is required by Bottom Avg 150.

So with Bottom Avg 150, reaching 280 is IMPOSSIBLE.

FIX: Change Bottom Avg to something lower? No, realistic electricity usage.

Change Overall Avg?

If Overall = 280. Total = 14000. Mid = 3000. Mid Avg = 300.

$4 y + 6 m + 21 = 3000 \Rightarrow 4 y + 6 m = 2979$ .

Min $y = 151 \Rightarrow 604$ .

$6 m = 2375 \Rightarrow m = 395$ . Too high.

Try Overall = 270. Total = 13500. Mid = 2500. Mid Avg = 250.

$4 y + 6 m = 2500 - 21 = 2479$ .

$6 m = 2479 - 604 = 1875$ .

$m = 312$ . Still high.

Try Overall = 264. Total = 13200. Mid = 2200. Mid Avg = 220.

$4 y + 6 m = 2200 - 21 = 2179$ .

$6 m = 2179 - 604 = 1575$ .

$m = 262.5$ .

This aligns perfectly with Option C (280 is close, maybe my y bound is loose?).

Wait, if $m = 262$ , then answer is 260.

Let's tweak Overall Avg to 266.

Total = 13300. Mid = 2300.

$4 y + 6 m = 2279$ .

$6 m = 2279 - 604 = 1675$ .

$m = 279.16$ .

Max integer $m = 279$ .

Closest option 280.

Decision: I will set Overall Average to 266 units.

This makes the theoretical max median ~279.

Option C (280) would be the distractor (just above max), Option B (270) the safe answer?

No, in "Maximum Possible" questions, the answer IS the boundary.

If max is 279, and options are 260, 270, 280, 290.

Then 270 is the max valid option? Or is 280 the intended answer implying my $y$ bound is softer?

Actually, $y = x_{21}$ . $x_{20}$ could be smaller if bottom set is skewed?

No, distinct integers avg 150. Min max-element is indeed ~160.

So $y \geq 161$ is safer.

If $y = 161 \Rightarrow 4 y = 644$ .

$6 m = 2279 - 644 = 1635 \Rightarrow m = 272.5$ .

Max integer $m = 272$ .

So with Overall=266, Max Median is 272.

Option B (270) is achievable. Option C (280) is not.

So Answer B is correct.

Final Parameter Set:

Top 20 Avg: 400

Bot 20 Avg: 150

Overall Avg: 266

Target Answer: 270 (as the closest valid option below the hard cap of 272).

Wait, "What is the maximum possible value?" usually expects the exact max.

I will provide options: 268, 270, 272, 274.

Correct: 272.

This is precise and exam-realistic.

Recalculating for final output verification:

Mid Sum = $50 (266) - 20 (400) - 20 (150) = 13300 - 8000 - 3000 = 2300$ .

Eq: $4 y + 6 m + 21 = 2300 \Rightarrow 4 y + 6 m = 2279$ .

Min $y$ : Bottom 20 distinct avg 150. Tightest packing ${141..160}$ . $x_{20} = 160$ .

$x_{21} \geq 161$ .

Max $m$ : Set $y = 161$ .

$6 m = 2279 - 644 = 1635$ .

$m = 272.5$ .

Max integer $m = 272$ .

Check validity: $x_{25} = 272, x_{26} = 273$ .

$x_{27..30} = 274, 275, 276, 277$ .

$x_{21..24} = 161, 162, 163, 164$ .

Sum = $650 + 545 + 1104 = 2299$ .

Need 2300. Add 1 to $x_{30} \to 278$ .

Valid.

Answer: 272.

Question 2 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

A car travels from City A to City B at an average speed of 60 km/h and returns from City B to City A at an average speed of 40 km/h. A second car travels the same round trip but maintains a constant speed of 50 km/h throughout. Which car completes the journey faster, and by what percentage of the FIRST car's total time?

A.
Second car is faster by 4%
B.
First car is faster by 4%
C.
Second car is faster by 8%
D.
Both take same time

Correct Answer:

Step-by-Step Solution

Key idea: This is a Harmonic vs Arithmetic Mean Trap disguised as a speed problem. Recognise it because equal distances with different speeds invoke Harmonic Mean, while constant speed invokes Arithmetic logic.

Step 1: Analyze Car 1 (Variable Speed).

Equal distance legs. Average Speed $\neq = \frac{60 + 40}{2} = 50$ .

True Avg Speed = Harmonic Mean = $\frac{2 \times 60 \times 40}{60 + 40} = \frac{4800}{100} = 48$ km/h.

Step 2: Analyze Car 2 (Constant Speed).

Avg Speed = 50 km/h.

Step 3: Compare Times.

Since Distance is same, Time $\propto \frac{1}{Speed}$ .

Higher Speed implies Lower Time.

Car 2 (50 km/h) > Car 1 (48 km/h). So Car 2 is faster.

Step 4: Calculate Percentage Difference relative to Car 1's Time.

Let Distance one-way = $D$ . Total = $2 D$ .

Time 1 = $\frac{2 D}{48}$ . Time 2 = $\frac{2 D}{50}$ .

Diff = $\frac{2 D}{48} - \frac{2 D}{50} = 2 D (\frac{1}{48} - \frac{1}{50}) = 2 D (\frac{50 - 48}{2400}) = \frac{4 D}{2400} = \frac{D}{600}$ .

% Faster = $\frac{Time Saved}{Original Time (Car 1)} \times 100$

$= \frac{D /600}{2 D /48} \times 100 = \frac{D}{600} \times \frac{48}{2 D} \times 100 = \frac{48}{1200} \times 100 = \frac{48}{12} = 4%$ .

Answer: Second car is faster by 4%.

Ratio, Proportion, Averages and Arithmetic Word Problems Short Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Summary: The Ratio & Mixture Toolkit

Summary: The Ratio & Mixture Toolkit

Summary: The Averages & Constraints Toolkit

Summary: The Averages & Constraints Toolkit

Summary: The Word Problems & Sets Toolkit

Summary: The Word Problems & Sets Toolkit

Ratio, Proportion, Averages and Arithmetic Word Problems: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

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