Data Sufficiency Data Insights Practice Questions

Let there be S Shovels and R rakes.
The wordy question language may then be simplified to reiterate the question as saying:
What is the value of (14S + 9R) ?


STATEMENT (1) alone: A 2 : 3 ratio does not give us a fix on the total number of shovels
and rakes purchased. It merely states the proportion of items that constitutes shovels and
rakes. For instance there can 2 Shovels and 3 Rakes or 4 Shovels and 6 Rakes or 8 Shovels
and 12 Rakes (and so on) in his purchase. Each possibility yields a different value for (14S +
9R). Which is why,

STATEMENT (1) alone - INSUFFICIENT


STATEMENT (2) alone: This statement says S + R = 50. Although this statement gives us a
fix on the total number purchased. It gives no clue as to the individual number of each
contained in the sum total of 50. Like the above explanation, we can have multiple sets of
(S,R) (say (10,40) or (25,25) for instance) values that add up to 50. All those values again
yield different value for (14S + 9R).

STATEMENT (2) alone - INSUFFICIENT


STATEMENT (1) & (2) together: Together we have a fix on both the total number and the
distribution (ratio) of the two items within the total. This is enough to yield a unique value for
both S & R. Alternatively, mathematically this may be seen as being given two equations: (1)
S = (2/3)*R & (2) S + R = 50 to solve for two variables: S & R uniquely. The unique set
(S,R) further yields a unique value of (14S + 9R). hence,

STATEMENT (1) & (2) together - SUFFICIENT

ANSWER – (C).

\(P\) is a positive odd integer → \(p\) can thus take on values \(\{1, 3, 5, 7, \ldots\}\)



STATEMENT (1) alone: \(p\) may thus be written as \(p = 8k + 5\) where \(k\) is a non-negative integer.

This means \((p - 5) = 8k\).

Or, \((p - 5)\) is divisible by 4 (since \((p - 5)\) is a multiple of 8).

Hence, \(\{(p - 5) + 4\} = (p - 1)\) is also divisible by 4.

Thus, \(p\) divided by 4 will always yield a remainder 1.

Unique solution.

STATEMENT (1) alone – SUFFICIENT



STATEMENT (2) alone: Since \(p\) is odd, if it is to be expressed as a sum of two positive integers then one of them must be even and the other odd.

Hence, the integers whose squares sum up to \(p\) are a pair of even and odd integers.

Mathematically this may be expressed as: \(p = (2k)^2 + (2m + 1)^2\); where \(k\) & \(m\) are non-negative integers.

Or, \(p = 4k^2 + 4m^2 + 4m + 1\)

Or, \(p = 4(k^2 + m^2 + m) + 1\)

Or, \(p = 4j + 1\)

The above divided by 4 will always yield a remainder 1.

Unique solution.

STATEMENT (2) alone – SUFFICIENT



ANSWER – (D).

The question introduces two variable sets with the possibility/certainty of an overlap. Such language is typical of two variable sets questions and these questions are best tackled by chalking out the information on a table.



Using the information given only in the question we can begin by creating our table and filling in the information and placing a '?' sign at the place that we're required to find.



$$\begin{array}{|c|c|c|c|} \hline & \text{Anti-lock brakes} & \text{No Anti-lock brakes} & \text{TOTAL} \\ \hline \text{Automatic transmission} & & ? & \\ \hline \text{No Automatic transmission} & & & \\ \hline \text{TOTAL} & & & 25 \\ \hline \end{array}$$



STATEMENT (1) alone: The additional information fills in the original table as follows:



$$\begin{array}{|c|c|c|c|} \hline & \text{Anti-lock brakes} & \text{No Anti-lock brakes} & \text{TOTAL} \\ \hline \text{Automatic transmission} & & ? & \\ \hline \text{No Automatic transmission} & 0 & & \\ \hline \text{TOTAL} & & & 25 \\ \hline \end{array}$$



This is too less information to even remotely arrive at anything. No unique value.

STATEMENT (1) alone - INSUFFICIENT



STATEMENT (2) alone: The additional information fills in the original table as follows:



$$\begin{array}{|c|c|c|c|} \hline & \text{Anti-lock brakes} & \text{No Anti-lock brakes} & \text{TOTAL} \\ \hline \text{Automatic transmission} & & ? & \\ \hline \text{No Automatic transmission} & & 2 & \\ \hline \text{TOTAL} & & & 25 \\ \hline \end{array}$$



Again, this is too less information to even remotely arrive at anything. No unique value.

STATEMENT (2) alone - INSUFFICIENT



STATEMENT (1) & (2) together: Both the statements together fill in the table completely as follows:



$$\begin{array}{|c|c|c|c|} \hline & \text{Anti-lock brakes} & \text{No Anti-lock brakes} & \text{TOTAL} \\ \hline \text{Automatic transmission} & ? & ? & 23 \\ \hline \text{No Automatic transmission} & 0 & 2 & 2 \\ \hline \text{TOTAL} & & & 25 \\ \hline \end{array}$$



Even clubbing the information in the 2 statements, the top two cells can still be filled with multiple pairs adding up to 23. No unique value.

STATEMENT (1) & (2) together - INSUFFICIENT

ANSWER – (E).