Question

In a \(1000 \mathrm{~m}\) race Amecsha gives a headstart of \(100 \mathrm{~m}\) to Bipasha and beats her by \(200 \mathrm{~m}\). In the same race Ameesha gives a headstart of \(100 \mathrm{~m}\) to Celina and beats her by \(300 \mathrm{~m}\). By how many metres would Bipasha beat Celina in a \(50 \mathrm{~m}\) race? (a) \(6.66 \mathrm{~m}\) (b) \(7.143 \mathrm{~m}\) (c) \(8 \mathrm{~m}\) (d) none of these

Short answer

a) 6.25 m b) 6 m c) 7.25 m d) none of these Answer: d) none of these Explanation: Using the given information, we calculated the distance covered by Bipasha and Celina in a 50-meter race to be 58.33 meters and 50 meters, respectively. Therefore, Bipasha would beat Celina by a distance of 8.33 meters, which is not among the given options.

Step-by-step solution

Find the ratio of distances covered by Amecsha and Bipasha in the 1000 m race

We know that Amecsha gives a headstart of 100 m to Bipasha and beats her by 200 m. This means that when Amecsha covers 1000 m, Bipasha covers (1000 - headstart - lead) meters, which is (1000 - 100 - 200) = 700 m. So, we have the ratio of distances covered: Amecsha : Bipasha = 1000 m : 700 m

Find the ratio of distances covered by Amecsha and Celina in the 1000 m race

We are also given that Amecsha gives a headstart of 100 m to Celina and beats her by 300 m. So, when Amecsha covers 1000 m, Celina covers (1000 - headstart - lead) meters, which is (1000 - 100 - 300) = 600 m. Now, we have the ratio of distances covered: Amecsha : Celina = 1000 m : 600 m

Find the ratio of distances covered by Bipasha and Celina

Combining the two ratios we found in steps 1 and 2, we get: Bipasha : Celina = (700 m / 1000 m) : (600 m / 1000 m) Simplifying the ratio, we get: Bipasha : Celina = 7 : 6

Find the distances covered by Bipasha and Celina in a 50 m race

If we assume Bipasha covers x meters in a 50 m race, with the ratio we found in step 3, we can write the equation: x / 50 = 7 / 6 Solving for x, we get: x = (7 * 50) / 6 = 58.33 m Since Bipasha covers 58.33 m in a 50 m race, Celina will cover 50 m.

Find the distance by which Bipasha beats Celina in a 50 m race

We can now calculate the distance by which Bipasha beats Celina in a 50 m race by taking the difference between the distances covered by Bipasha and Celina: Distance = 58.33 m - 50 m = 8.33 m The answer is not found among the given options, so the correct answer is (d) none of these.

Key concepts

Quantitative Aptitude

When we talk about quantitative aptitude, we refer to the ability to handle numerical and logical reasoning problems. It involves the application of mathematical concepts and techniques to solve real-world problems or puzzles, such as the race problem from our exercise.

In this context, quantitative aptitude encompasses various mathematical skills, including ratio and proportion, arithmetic calculations, and understanding of distance and speed. Given that many competitive exams and selection processes test this skill, it’s essential to approach such problems with a logical mindset and systematic problem-solving technique.

Ratio and Proportion

The concept of ratio and proportion is a fundamental aspect of mathematics that deals with the comparison of quantities and their relative sizes. Ratios are expressed as two or more numbers that reflect how many times one number contains another. Proportions, on the other hand, assert that two ratios are equal.

These are instrumental in solving problems like the one involving the race, where understanding the proportional relationship between the distances covered by different athletes allows us to determine their relative performances. It's critical to identify and maintain the consistency of units when dealing with ratios and proportions to ensure accurate calculations.

Distance and Speed Calculations

Calculating distance and speed is central to solving many problems in quantitative aptitude. The basic formula you’d use is 'Distance = Speed × Time.' However, these problems can become more complex when they involve comparative speeds, variable speeds, or, as in our exercise, a head start.

In the context of our race problem, we're using the concepts of ratio and speed to calculate the relative distances covered by each athlete. By establishing a ratio, we can project distances covered over different lengths, like scaling down from a 1000 m race to a 50 m one. Always remember to check the congruency of the units you're working with, so meters stay with meters, and seconds with seconds, to avoid any mismatch in your calculations.