Fisher's index is the geometric mean of Laspeyre's and Paasche's indices: $P_{01}(F)=\sqrt{P_{01}(L)\cdot P_{01}(P)} = 180$.

Fisher's ideal price index number is defined as the geometric mean of Laspeyre's index $P_{01}(L)$ and Paasche's index $P_{01}(P)$:

$P_{01}(F) = \sqrt{P_{01}(L)\times P_{01}(P)}$

Substituting the given values $P_{01}(L)=225$ and $P_{01}(P)=144$:

$P_{01}(F) = \sqrt{225\times 144} = \sqrt{32400}$

Since $\sqrt{225}=15$ and $\sqrt{144}=12$, we get $\sqrt{32400}=15\times 12 = 180$.

Dorbish–Bowley's index is the arithmetic mean of Laspeyre's and Paasche's indices: $P_{01}(D\!-\!B)=\dfrac{P_{01}(L)+P_{01}(P)}{2}=\dfrac{90+40}{2}=65$.

Formula. The Dorbish–Bowley price index number is defined as the arithmetic mean of the Laspeyre's index $P_{01}(L)$ and the Paasche's index $P_{01}(P)$:

$P_{01}(D\!-\!B) = \frac{P_{01}(L) + P_{01}(P)}{2}.$

Substitution. Given $P_{01}(L)=90$ and $P_{01}(P)=40$,

$P_{01}(D\!-\!B) = \frac{90 + 40}{2} = \frac{130}{2} = 65.$

A weighted quantity index measures how total quantity has changed from the base year to the current year, giving each commodity a weight $w$. The formula is $Q_{01} = \dfrac{\sum q_1 w}{\sum q_0 w} \times 100$.

In the simple aggregate quantity index we would just add the quantities. But different commodities matter to different degrees, so we attach a weight $w$ (commonly the price, or value) to each item.

For each commodity we form the weighted quantity of the current year, $q_1 w$, and of the base year, $q_0 w$. Summing over all commodities gives $\sum q_1 w$ and $\sum q_0 w$.

The index expresses the current-year total as a percentage of the base-year total:

$Q_{01} = \frac{\sum q_1 w}{\sum q_0 w} \times 100.$

Dorbish–Bowley's index is the average of Laspeyre's and Paasche's indices: $P_{01}^{DB}=\dfrac{1}{2}\left(\dfrac{\sum p_1 q_0}{\sum p_0 q_0} + \dfrac{\sum p_1 q_1}{\sum p_0 q_1}\right)\times 100$ — option (c).

The two base index numbers are:

$\text{Laspeyre's: } P_{01}^{L}=\frac{\sum p_1 q_0}{\sum p_0 q_0}\times 100 \qquad \text{(base-year quantity weights)}$

$\text{Paasche's: } P_{01}^{P}=\frac{\sum p_1 q_1}{\sum p_0 q_1}\times 100 \qquad \text{(current-year quantity weights)}$

Dorbish–Bowley's price index number is defined as their arithmetic mean:

$P_{01}^{DB}=\frac{P_{01}^{L}+P_{01}^{P}}{2}=\frac{\dfrac{\sum p_1 q_0}{\sum p_0 q_0} + \dfrac{\sum p_1 q_1}{\sum p_0 q_1}}{2}\times 100$

Comparing with the options, this matches option (c). (Option (b) is the form of Fisher-type reciprocals, and options (a) and (d) use incorrect ratio pairings.)

The weighted aggregate quantity index number is the ratio of the aggregate of weighted current-year quantities to weighted base-year quantities, $\dfrac{\sum q_1 w}{\sum q_0 w}\times 100$. Hence option (C).

For a quantity index number by the weighted aggregate method, each quantity is multiplied by its weight $w$ and the aggregates are compared, with the current year in the numerator and the base year in the denominator:

$Q_{01} = \dfrac{\sum q_1 w}{\sum q_0 w} \times 100$

Here $q_1$ = current-year quantities, $q_0$ = base-year quantities, and $w$ = weights. The options that sum each ratio term separately, $\sum\frac{q_1 w}{q_0 w}$, are wrong (that would be an average-of-relatives form, not the aggregate form), and putting $q_0$ on top would reverse base and current years.

Laspeyre's price index weights prices by base-year quantities, $P_{01}^{L} = \dfrac{\sum p_1 q_0}{\sum p_0 q_0}\times 100$; current-year quantities are used by Paasche's index. Hence the statement is False.

The two weighted-aggregate price index numbers differ only in the choice of quantity weights:

$\text{Laspeyre: } P_{01}^{L} = \dfrac{\sum p_1 q_0}{\sum p_0 q_0}\times 100 \quad(\text{base-year quantities } q_0 \text{ as weights})$

$\text{Paasche: } P_{01}^{P} = \dfrac{\sum p_1 q_1}{\sum p_0 q_1}\times 100 \quad(\text{current-year quantities } q_1 \text{ as weights})$

Since Laspeyre's index fixes the weights at the base-year quantities $q_0$ (not the current-year quantities $q_1$), the claim in the statement is wrong.

Walsh's price index uses the geometric mean of the two quantities as weight: $P_{01} = \dfrac{\sum p_1 \sqrt{q_0 q_1}}{\sum p_0 \sqrt{q_0 q_1}} \times 100$.

In a weighted aggregate index number, each item's price is weighted by a suitable quantity. Walsh's method takes the weight of each commodity as the geometric mean of the base-year quantity $q_0$ and the current-year quantity $q_1$, i.e. $\sqrt{q_0 q_1}$.

With $p_0, p_1$ the base-year and current-year prices, the current-year prices are aggregated (weighted by $\sqrt{q_0 q_1}$) and divided by the similarly weighted base-year prices:

$P_{01} = \frac{\sum p_1 \sqrt{q_0 q_1}}{\sum p_0 \sqrt{q_0 q_1}} \times 100$

Using the geometric mean of quantities makes Walsh's index a good compromise between Laspeyre's ($q_0$ weights) and Paasche's ($q_1$ weights) indices.

The weighted aggregate price index is the ratio of total weighted current-year prices to total weighted base-year prices, times $100$.

Let $p_0$ and $p_1$ be the base-year and current-year prices and $w$ the assigned weights. The Price Index Number by the Weighted Aggregate Method aggregates the weighted prices and compares the current year with the base year:

$P_{01} = \dfrac{\sum p_1 w}{\sum p_0 w}\times 100$.

The numerator and denominator must each be a sum over all commodities, which rules out the options without both summation signs; and the current year ($p_1$) belongs in the numerator, which rules out the option with $p_0 w$ on top.

Dorbish-Bowley's index is the arithmetic mean of Laspeyre's and Paasche's; the geometric mean (square root of their product) is Fisher's index — so the statement is False.

Let $L$ and $P$ denote Laspeyre's and Paasche's price index numbers.

Dorbish-Bowley's Price Index Number is their arithmetic mean:

$\text{DB} = \dfrac{L + P}{2}$.

Fisher's Ideal Price Index Number is their geometric mean:

$\text{Fisher} = \sqrt{L \cdot P}$.

The statement describes $\sqrt{L\cdot P}$, which is Fisher's index — not Dorbish-Bowley's. Hence the statement is incorrect.

Fisher's ideal price index is the geometric mean of the Laspeyres and Paasche indices, which gives $P_{01}(F)=110$.

Fisher's ideal price index number is defined as the geometric mean of the Laspeyres index $P_{01}(L)$ and the Paasche index $P_{01}(P)$:

$P_{01}(F) = \sqrt{P_{01}(L)\times P_{01}(P)}$

Substituting the given values $P_{01}(L)=121$ and $P_{01}(P)=100$:

$P_{01}(F) = \sqrt{121\times 100} = \sqrt{12100} = 110$