Which is faster between the tiger and the leopard given the relation between the number of paces they take and length of their paces?

Question

$I.$ For every $4$ paces taken by the tiger, the leopard takes $5$ paces.

$II.$ The total length of $10$ paces of the tiger is less than the total length of $9$ paces of the leopard.

The answer can be found using:

$A.$ Only statement I and not II

$B.$ Only statement II and not I

$C.$ Both I and II together, but not using any of them alone

$D.$ Either I or II

$E.$ None of them (even both of them together are not sufficient)

Following is my solution:

Using $I:$ Let the length of a pace taken by the tiger = $x$ & length of a pace taken by the leopard = $y$

Given that $4x = 5y$, i.e. at the same time, the tiger covers a length of $4x$ and the leopard covers a length of $5y$. Since we don't know the values of $x$ and $y$, thus we can't find which one is faster. Hence I is not sufficient.

$Using II:$ Let the length of a pace taken by the tiger = $x$ & length of a pace taken by the leopard = $y$

Given that $10x < 9y$ which means $x < 9y/10$

Since we don't know the values of $x$ and $y$, thus we can't find which one is faster. Hence II is not sufficient.

Combining $I$ & $II:$ From $I$, we have: $4x = 5y$ -------1)

From $II$, we have: $x < 9y/10$ ------2)

Substituting the maximum value of $2)$ in $1)$, we have:

$4*9y/10 < 5y$

$3.6y < 5y$

Going by this, can I not conclude that the leopard is faster than the tiger? Also, according to the $IInd$ statement, we can say that the length of each pace of the tiger is less than the length of each pace of the leopard. According to $Ist$ statement, the tiger takes $4$ paces and the leopard takes $5$ paces in the same amount of time. Thus, combining both statements, we can say that in the same amount of time, the length covered by the tiger is less than the length covered by the leopard. Hence, the leopard is faster.

However, the answer given in my booklet is option $E$ - that even both statements together are not sufficient. The following image shows the solution given in the booklet:

Can someone explain this solution to me, and also, why I am wrong in my approach?

Answer 1

Your answer is correct (based on the paragraph where you explain the result in words). The book's solution is wrong.

The book makes at least two mistakes. It says that $\frac54$ paces of the leopard covers a distance less than $\frac{9}{10}x \cdot \frac54$ meters.

But the pace of the leopard was already defined to be $x$ meters exactly. Therefore $\frac54$ paces of the leopard covers a distance of $\frac54 x$ meters. (Mistake number one.) And that's exactly $\frac54 x$ meters, not "less than" as your book says. (Mistake number two.)

It is the pace of the tiger which must be less than $\frac{9}{10}x $ meters, given that the leopard's pace is $x$ meters. And indeed $\frac{9}{10}x < \frac54 x$ no matter what the value of $x$ is (provided that $x$ is positive, of course).

It appears that in the middle of writing the book's answer, whoever wrote it got confused about which was the tiger and which was the leopard.

---

I would use a bit of caution when expressing your arguments in formulas. According to your assignment of variables, the tiger travels the distance $4x$ in the same amount of time that the leopard travels the distance $5y.$ So there is a correspondence between $4x$ and $5y,$ but the two are not equal. As you observed, $x<\frac{9}{10}y.$ This implies that $4x<3.6y,$ and (as you also observed) $3.6y<5y.$ What you did not exactly say (but you appear to have been thinking) is that the two inequalities chain together to give $4x<5y,$ thereby proving that the leopard travels farther.

Answer 2

I agree with David K that the book is wrong.

We can avoid both fractions and algebra (and thus reduce the opportunity for committing silly mistakes) by letting the animals run for a longer period of time -- for example five times the situation described in (I).

Then the tiger has run $20$ tiger paces while the leopard has run $25$ leopard paces.

By (II), the tiger's distance is less than $18$ leopard paces, and therefore certainly shorter than the $25$ leopard paces the leopard has run. So the leopard is definitely faster.

---

Or, for that matter, entirely without numbers. Derive from each of the givens:

Ia: The tiger takes fewer paces than the leopard.

IIa: A tiger pace is shorter than a leopard pace.

Answer 3

Alternatively, as you denoted, let $x$ and $y$ be the lengths of a unit pace of tiger and leopard, respectively.

I. For every 4 paces taken by the tiger, the leopard takes 5 paces.

It does not imply $4x=5y$, but $4x\sim 5y$. Which distance is greater depends on the values of $x$ and $y$. Also, it implies: $\color{red}{10x}\sim\color{blue}{12.5y}$.

II. The total length of 10 paces of the tiger is less than the total length of 9 paces of the leopard.

It implies: $10x<9y$. Now, using the first condition: $\color{red}{10x}<9y<\color{blue}{12.5y}, x>0,y>0$.

Hence, the leopard goes longer distance, therefore it is faster.