UG Applied Mathematics Booster Test 2 - Methods of Index Numbers -

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UG Applied Mathematics Booster Test 2 - Methods of Index Numbers

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

UG Applied Mathematics Booster Test 2 - Methods of Index Numbers

1 / 20

QUESTION 1 OF 20

Calculate the simple aggregative price summation index for the year 2008 (base year 2000), given that the unit prices in 2000 are
"₹" 10," ₹" 20,
and in 2008 are
"₹" 15," ₹" 25.
"Simple Aggregative Price Index"=(∑p_n)/(∑p_0 )×100

Answer Review

1 Calculate the simple aggregative price summation index for the year 2008 (base year 2000), given that the unit prices in 2000 are
"₹" 10," ₹" 20,
and in 2008 are
"₹" 15," ₹" 25.
"Simple Aggregative Price Index"=(∑p_n)/(∑p_0 )×100

A 120.00

B 133.33

C 140.00

D 150.00

💡 Short Explanation

Add current year prices Add base year prices Apply simple aggregative formula

✅ Correct Answer Explanation

The Simple Aggregative Price Index formula is: \(P_{01}=\frac{\sum p_{n}}{\sum p_{0}}\times 100\) Base year total: \(10+20=30\) Current year total: \(15+25=40\) Thus, \(\frac{40}{30}\times 100=133.33\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Represents a smaller increase than calculated.
Option C → Incorrect arithmetic evaluation.
Option D → Would require current total to be 45.

🎯 Strategy

Used

Substitution

Application:

Direct substitution into the aggregative index formula gives the answer quickly.

Final Logic:

\(40\div 30\times 100=133.33\)

🧠 Memory Trick

"Current sum ÷ Base sum ×100"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Simple Aggregative Method –

2 Which of the following is an incorrect statement regarding quantity summation?

A It is known as an unweighted aggregate approach.

B The index number is (∑Q_n)/(∑Q_0 )×100.

C It incorporates base and current prices in its formula.

D It measures the collective change in a group of items.

💡 Short Explanation

Quantity index uses quantities only Prices are not included It is an unweighted approach

✅ Correct Answer Explanation

A quantity summation index focuses only on quantities: \(Q_{01}=\frac{\sum Q_{n}}{\sum Q_{0}}\times 100\) It does not include prices in the formula. Therefore, Option C is incorrect. Options A, B, and D correctly describe quantity summation.

❓ Why Other Options Are Incorrect

Option A → Correct because simple aggregation is unweighted.
Option B → Correct standard quantity index formula.
Option D → Correct since it measures group change collectively.

🎯 Strategy

Used

Elimination

Application:

Identify the only statement introducing prices into a quantity formula.

Final Logic:

Quantity index excludes price variables.

🧠 Memory Trick

"Quantity index = Quantities only"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Quantity Index –

3 The weight absence problem signifies that the major flaw in using absolute quantities without relatives in simple aggregative methods is:

A It is mathematically impossible to sum.

B It applies arbitrary geometric means.

C It ignores all prices entirely.

D It is heavily influenced by commodities with large per unit pricing.

💡 Short Explanation

No explicit weights are assigned High-priced items dominate totals Index becomes biased

✅ Correct Answer Explanation

Simple aggregative methods add raw prices directly. Commodities with very high prices automatically get more influence. Thus, expensive commodities act as concealed weights, causing bias in the index. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Summation is mathematically possible.
Option B → Geometric means are not used here.
Option C → Prices are central to the method.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Link "weight absence" with domination by costly commodities.

Final Logic:

No weights means costly items dominate automatically.

🧠 Memory Trick

"Costly goods control unweighted sums"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Limitation of Simple Aggregative Method –

4 Assertion (A): The simple aggregative method prevents high value bias effectively.
Reason (R): It uses varying weights for quantities mathematically to smooth out differences.

A Both A and R are false

B A is true, R is false

C Both A and R are true, and R is the correct explanation of A

D A is false, R is true

💡 Short Explanation

Simple aggregation has bias No varying weights are used Both statements are incorrect

✅ Correct Answer Explanation

The Assertion is false because simple aggregative methods actually suffer from high-value bias. The Reason is also false because no varying weights are used in simple aggregation. Therefore, both A and R are false.

❓ Why Other Options Are Incorrect

Option B → Assertion itself is false.
Option C → Neither statement is true.
Option D → Reason is also incorrect.

🎯 Strategy

Used

Elimination

Application:

Recall that simple aggregation is unweighted and biased.

Final Logic:

Both the assertion and reason contradict the method's limitation.

🧠 Memory Trick

"Simple aggregate = No weights, more bias"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Defects of Aggregative Method –

5 Representing base price and current price as scalar components of a vector, the relative conversion for a specific commodity iis calculated as:

A (p_0i/p_ni )×100

B (p_ni/p_0i )×100

C (p_ni×p_0i)/100

D (p_ni-p_0i)×100

💡 Short Explanation

Relative compares current to base Multiply by 100 for percentage Standard price relative formula

✅ Correct Answer Explanation

The price relative formula is: \(R_{i}=\frac{p_{ni}}{p_{0i}}\times 100\) It measures the percentage change in current price relative to the base year. Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Inverse formula.
Option C → Incorrect multiplication operation.
Option D → Gives absolute change, not relative percentage.

🎯 Strategy

Used

Formula Recall

Application:

Identify the standard price relative expression.

Final Logic:

Current price divided by base price gives relative movement.

🧠 Memory Trick

"Relative = Current over Base"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Price Relatives –

6 If the continuous integral of price variations is simplified into a discrete mean calculation of relatives for nitems, the formula used is:

A (∑p_n)/(∑p_0 )

B ∫(p_n Q_n)" " dp

C 1/n∑(p_n/p_0 ⓜ×100)

D ∑(p_0 Q_0)

💡 Short Explanation

Relatives are averaged arithmetically Divide total relatives by number of items Produces average relative index

✅ Correct Answer Explanation

Simple Average of Price Relatives is: \(P=\frac{1}{n}\sum \left(\frac{p_{n}}{p_{0}},\ 100\right)\) This computes the arithmetic mean of all price relatives. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Simple aggregative formula, not average relatives.
Option B → Irrelevant continuous integral form.
Option D → Only represents base-year value totals.

🎯 Strategy

Used

Option Grouping

Application:

Identify the only option averaging relative percentages.

Final Logic:

Average relatives require summation divided by number of commodities.

🧠 Memory Trick

"Add relatives, divide by n"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Simple Average of Relatives –

7 Arrange the indices logically from the one with the biggest equal-importance flaw, to the one that uses current weights, to the geometric ideal:
1. Simple Average of Relatives
2. Paasche Index
3. Fisher's Ideal Index
4. Value Index

A 4, 3, 2, 1

B 1, 3, 2, 4

C 2, 1, 3, 4

D 1, 2, 3, 4

💡 Short Explanation

Simple average gives equal importance Paasche uses current weights Fisher uses geometric mean

✅ Correct Answer Explanation

Logical progression: 1. Simple Average of Relatives → equal-importance flaw 2. Paasche → current-period weights 3. Fisher → geometric ideal method 4. Value Index → combines price and quantity value comparison Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect conceptual sequence.
Option B → Fisher should come after Paasche.
Option C → Starts incorrectly with Paasche.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Arrange methods according to conceptual sophistication.

Final Logic:

Equal-weight method precedes weighted and ideal methods.

🧠 Memory Trick

"Simple → Weighted → Ideal"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Types of Index Methods –

8 Match the methods with their approach to ignoring quantities or using weights.

List I (Methods)	List II (Approach)
1. Simple Average of Relatives Method	a. Disregards absolute quantities, giving equal importance to each price relative
2. Laspeyres Index Method	b. Uses base period quantities as fixed weights
3. Paasche Index Method	c. Uses current period quantities as weights
4. Fisher's Ideal Index Method	d. Uses the geometric mean of Laspeyres and Paasche indices

A 1–a, 2–b, 3–c, 4–d

B 1–b, 2–a, 3–d, 4–c

C 1–c, 2–d, 3–a, 4–b

D 1–d, 2–c, 3–b, 4–a

💡 Short Explanation

Laspeyres uses base weights Paasche uses current weights Fisher combines both geometrically

✅ Correct Answer Explanation

Correct matching: • Simple Average → equal importance → a • Laspeyres → base quantities → b • Paasche → current quantities → c • Fisher → geometric mean → d Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Swaps Laspeyres and Simple Average.
Option C → Completely mismatched pairings.
Option D → Incorrectly reverses several methods.

🎯 Strategy

Used

Option Grouping

Application:

Recall signature characteristic of each index method.

Final Logic:

Each method is identified by its weighting system.

🧠 Memory Trick

"L = Last year weights, P = Present weights"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Weighted Index Methods –

9 In a bar chart representing expenditures, the area of each bar is given by P×Q. Using base quantities as weights, calculate the weighted aggregative index if
∑p_0 Q_0=100"and"∑p_n Q_0=150.
"Weighted Aggregative Index"=(∑p_n Q_0)/(∑p_0 Q_0 )×100

A 100

B 150

C 250

D 125

💡 Short Explanation

Use weighted aggregative formula Divide current weighted total by base weighted total Multiply by 100

✅ Correct Answer Explanation

Weighted Aggregative Index: \(P=\frac{\sum p_{n}Q_{0}}{\sum p_{0}Q_{0}}\times 100\) Substitute values: \(\frac{150}{100}\times 100=150\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Indicates no change.
Option C → Incorrect multiplication.
Option D → Incorrect ratio evaluation.

🎯 Strategy

Used

Substitution

Application:

Direct insertion into formula simplifies calculation.

Final Logic:

\(150\div 100\times 100=150\)

🧠 Memory Trick

"Weighted index = Weighted current over weighted base"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Weighted Aggregative Method –

10 The probability of an index reflecting true economic conditions increases when quantity importance is considered. A weighted aggregative index achieves this by:

A Taking the simple mean of prices

B Using relative prices only

C Assigning appropriate quantity weights to different commodities

D Disregarding consumption units

💡 Short Explanation

Important goods receive higher weight Consumption significance is reflected Index becomes more realistic

✅ Correct Answer Explanation

Weighted aggregative indices assign suitable quantity weights to commodities according to their importance. This improves realism and economic accuracy because heavily consumed goods affect the index more strongly. Therefore, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Simple means ignore importance differences.
Option B → Relative prices alone are insufficient.
Option D → Consumption units are essential for weighting.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Link "true economic conditions" with realistic weighting.

Final Logic:

Proper quantity weights improve representativeness.

🧠 Memory Trick

"More used goods = More weight"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Importance of Weighted Indices –

📄 Passage

"In 1874, German statistician Paasche suggested that for determining weights, one should revise the quantities over time. When fixed-quantity weights are considered from current period usage, the resulting weighted aggregative index is known as the Paasche Index. The construction of the Paasche index requires a new set of weights for each year under consideration."

Based on the contrasting methods in the passage, if one were to use base year weights (Laspeyres method) for data where
∑p_n Q_0=110"and"∑p_0 Q_0=80,
the value of the index is:
"Laspeyres Index"=(∑p_n Q_0)/(∑p_0 Q_0 )×100

A 100.0

B 120.5

C 130.0

D 137.5

💡 Short Explanation

Laspeyres uses base year quantities Substitute values directly into formula Weighted price increase equals 137.5

✅ Correct Answer Explanation

Laspeyres Price Index is calculated using: \(L=\frac{\sum p_{n}Q_{0}}{\sum p_{0}Q_{0}}\times 100\) Substituting the given values: \(L=\frac{110}{80}\times 100L=1.375\times 100=137.5\) Hence, Option D is correct. Option A is incorrect because no change would require equal numerator and denominator. Option B and C are incorrect numerical evaluations.

❓ Why Other Options Are Incorrect

Option A → Represents no price increase, which is not true here.
Option B → Incorrect arithmetic calculation.
Option C → Does not match the proper ratio calculation.

🎯 Strategy

Used

Substitution

Application:

Directly place the numerical values into the Laspeyres formula.

Final Logic:

\(110\div 80\times 100=137.5\)

🧠 Memory Trick

"Laspeyres = Old quantities as weights"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Laspeyres Price Index –

📄 Passage

Assertion (A): The Laspeyres index always completely underestimates the rise in living costs.
Reason (R): The Laspeyres method uses current year quantities as weights.

A Both A and R are false

B A is true, R is false

C Both A and R are true, and R is the correct explanation of A

D A is false, R is true

💡 Short Explanation

Laspeyres generally overestimates inflation It uses base year quantities Both statements are incorrect

✅ Correct Answer Explanation

The Assertion is false because Laspeyres Index tends to overestimate the rise in cost of living, not underestimate it. The Reason is also false because Laspeyres uses base-year quantities \(Q_{0}\), not current-year quantities. Therefore, both A and R are false, making Option A correct.

❓ Why Other Options Are Incorrect

Option B → Assertion itself is incorrect.
Option C → Both statements are not true.
Option D → Reason is also false because current-year quantities belong to Paasche's method.

🎯 Strategy

Used

Elimination

Application:

Recall that Laspeyres uses base weights and overestimates inflation.

Final Logic:

Both the assertion and reason contradict the standard concept.

🧠 Memory Trick

"Laspeyres = Base weights + Higher inflation estimate"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Limitation of Laspeyres Index –

13 Unlike a smoothing moving average, the Paasche index uses discrete current year weights. What is its mathematical formula?

A (∑p_n Q_0)/(∑p_0 Q_0 )×100

B (∑p_n Q_n)/(∑p_0 Q_n )×100

C (∑p_0 Q_0)/(∑p_n Q_n )×100

D (∑p_n (Q_0+Q_n))/(∑p_0 (Q_0+Q_n))×100

💡 Short Explanation

Paasche uses current quantities Current weights reflect recent consumption It is a weighted aggregative index

✅ Correct Answer Explanation

Paasche Price Index uses current-period quantities as weights. Formula: \(P=\frac{\sum p_{n}Q_{n}}{\sum p_{0}Q_{n}}\times 100\) Thus, Option B is correct. Option A represents Laspeyres Index. Option D represents Marshall–Edgeworth Index.

❓ Why Other Options Are Incorrect

Option A → Uses base-year quantities, so it is Laspeyres' method.
Option C → Formula structure is mathematically reversed.
Option D → Uses combined weights from both years.

🎯 Strategy

Used

Option Grouping

Application:

Identify the formula using current-year weights \(Q_{n}\).

Final Logic:

Paasche always uses current quantities.

🧠 Memory Trick

"Paasche Prefers Present quantities"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Paasche Index –

14 If plotted on a graph representing inflation regions, the Paasche curve often lies below the Laspeyres curve. Why does this underestimation occur?

A It strictly assumes all prices remain static over time

B It uses geometric calculations implicitly

C Calculations are based on current period weights, adjusting to items that have become proportionately cheaper

D It relies exclusively on base year volume

💡 Short Explanation

Current quantities reflect substitution Expensive items receive lower weights Inflation estimate becomes smaller

✅ Correct Answer Explanation

Paasche Index uses current-period quantities. Consumers generally reduce consumption of costly goods and purchase relatively cheaper substitutes. Thus, expensive goods receive lower importance, causing the index to underestimate the rise in living costs. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Paasche does not assume static prices.
Option B → Geometric calculations relate to Fisher's Index.
Option D → Base-year volume belongs to Laspeyres' method.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Connect current weights with consumer substitution behavior.

Final Logic:

Current weights reduce measured inflation.

🧠 Memory Trick

"Present weights → Lower inflation"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Limitation of Paasche Method –

15 If the Laspeyres index yields 144and the Paasche index yields 100, then using the geometric mean concept, the Fisher Index is given by:
"Fisher Index"=√("Laspeyres" ×"Paasche" )

A 122

B 244

C 144

D 120

💡 Short Explanation

Fisher uses geometric mean Multiply Laspeyres and Paasche Take square root of product

✅ Correct Answer Explanation

Fisher's Ideal Index formula is: \(F=\sqrt{LP}\) Given: \(L=144,P=100F=\sqrt{144\times 100}F=\sqrt{14400}=120\) Therefore, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect square root evaluation.
Option B → Product taken without square root.
Option C → Equal to Laspeyres only, not Fisher.

🎯 Strategy

Used

Substitution

Application:

Directly substitute the values into Fisher's formula.

Final Logic:

\(\sqrt{14400}=120\)

🧠 Memory Trick

"Fisher Finds √LP"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Fisher's Ideal Index –

16 To establish the ideal index concept, Fisher's index satisfies the time-reversal test. Which mathematical condition defines this test?

A P_01×P_10=1

B P_01×P_10>1

C P_01×P_10<1

D P_01×P_10=0

💡 Short Explanation

Time reversal checks consistency Forward and backward indices are reciprocals Product must equal unity

✅ Correct Answer Explanation

The Time Reversal Test states: \(P_{01}\times P_{10}=1\) This means the index from period 0 to 1 multiplied by the reverse index must equal 1. Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Product greater than 1 violates reversal consistency.
Option C → Product less than 1 also violates the condition.
Option D → Product cannot be zero for valid indices.

🎯 Strategy

Used

Memory Recall / Elimination

Application:

Recall the standard adequacy condition for Fisher's Index.

Final Logic:

Reciprocal indices multiply to 1.

🧠 Memory Trick

"Reverse twice → Return to 1"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Time Reversal Test –

17 The average weights method in the Marshall–Edgeworth formulation relies on combining:

A Only base quantities

B Average of base and current year quantities (Q_0+Q_n)/2

C Only current period prices

D Simple price summation

💡 Short Explanation

Both year quantities are averaged Average weights reduce bias Combines old and new consumption patterns

✅ Correct Answer Explanation

Marshall–Edgeworth Index uses the average of base-year and current-year quantities as weights. The average weight is: \(\frac{Q_{0}+Q_{n}}{2}\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Uses only base quantities, which is Laspeyres' idea.
Option C → Prices alone are not weights.
Option D → Simple summation ignores weighting.

🎯 Strategy

Used

Option Grouping

Application:

Identify the option involving combined quantities.

Final Logic:

Marshall–Edgeworth averages both period quantities.

🧠 Memory Trick

"Marshall mixes both years"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Marshall–Edgeworth Method –

18 In analyzing combined period weights, if current quantities perfectly match base quantities (Q_0ⓜ=Q_n ), what happens to the Marshall–Edgeworth index?

A It fails and becomes undefined

B It doubles in output value

C It reduces to the simple weighted aggregative index identical to Laspeyres and Paasche

D It mathematically becomes zero

💡 Short Explanation

Equal quantities remove weighting differences Laspeyres and Paasche become equal Marshall–Edgeworth also becomes identical

✅ Correct Answer Explanation

If: \(Q_{0}=Q_{n}\) then all three methods use identical weights. Thus: • Laspeyres = Paasche • Marshall–Edgeworth also gives the same result Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Formula remains valid.
Option B → No doubling occurs mathematically.
Option D → Index cannot become zero due to equal quantities.

🎯 Strategy

Used

Substitution

Application:

Replace \(Q_{0}\)with \(Q_{n}\)conceptually in all formulas.

Final Logic:

Equal weights produce identical weighted indices.

🧠 Memory Trick

"Equal quantities = Equal indices"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Marshall–Edgeworth Method –

19 When employing the relative with weights method, the specific value weights utilized for the base period in the formula
(∑[(p_n/p_0 )ⓜ×V])/(∑V)
are denoted by:

A p_n Q_n

B Q_0

C p_n

D p_0 Q_0

💡 Short Explanation

Base-year value weights are used Value = Price × Quantity Weights improve representativeness

✅ Correct Answer Explanation

In weighted average of relatives, the value weights are generally: \(V=p_{0}Q_{0}\) This represents base-year value weights. Therefore, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Represents current-year value, not base value.
Option B → Quantity alone is incomplete.
Option C → Price alone cannot act as value weight.

🎯 Strategy

Used

Dimensional/Unit Analysis

Application:

Value weights must combine price and quantity.

Final Logic:

Value = Price × Quantity = \(p_{0}Q_{0}\)

🧠 Memory Trick

"Value weight = Price × Quantity"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Weighted Average of Relatives –

20 Which of the following is incorrect regarding the improved accuracy of the weighted average of relatives?

A The index number calculation highly depends on the units in which the prices are quoted

B It allows assigning varying weights, making commodities compatible for summation

C It calculates identical results to the Laspeyres index when p_0 Q_0is used as weights

D It overcomes the equal-importance flaw of simple averages

💡 Short Explanation

Weighted relatives reduce unit problems Weights improve comparability Equal-importance flaw is removed

✅ Correct Answer Explanation

Weighted Average of Relatives is less affected by units because it uses relatives (percentages), not raw price totals. Thus, saying the calculation "highly depends on units" is incorrect. Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Correct because weights improve compatibility.
Option C → Using \(p_{0}Q_{0}\)weights gives results close to Laspeyres.
Option D → Weighted methods correct the equal-importance issue.

🎯 Strategy

Used

Extreme Word Filter

Application:

The phrase "highly depends" signals conceptual inaccuracy.

Final Logic:

Relative methods reduce dependence on measurement units.

🧠 Memory Trick

"Relatives reduce unit effect"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Weighted Average of Relatives –

UG Applied Mathematics Booster Test 2 - Methods of Index Numbers

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