Arjun And Naina's Soft Drink Problem Solving Fractions
This article delves into a mathematical problem involving Arjun and Naina, who share two bottles of soft drink. We will explore how much soft drink they consumed and what fraction of the total quantity they drank. This problem provides a practical application of fractions and helps illustrate how mathematical concepts are used in everyday situations. Understanding fractions is crucial for various real-life scenarios, from cooking and baking to managing finances and understanding proportions.
Problem Statement
Arjun and Naina each have a bottle containing 2 liters of soft drink. They both drink 2/5 of the soft drink from their respective bottles, and they keep the remaining amount for later. Our task is to determine:
- How much soft drink did both of them drink in total?
- What fraction of the total quantity of soft drink did they drink?
This problem requires us to calculate the amount of soft drink consumed by each person and then combine those amounts to find the total. Additionally, we need to express this total consumption as a fraction of the overall quantity of soft drink they initially had.
Part A: Calculating the Total Soft Drink Consumed
Step 1: Soft drink consumed by Arjun
To determine the amount of soft drink Arjun drank, we need to calculate 2/5 of the 2 liters in his bottle. This involves multiplying the fraction 2/5 by the quantity 2 liters. The calculation is as follows:
(2/5) * 2 liters = 4/5 liters
Thus, Arjun drank 4/5 liters of soft drink. This step highlights the fundamental operation of multiplying a fraction by a whole number, a key concept in understanding proportions and ratios. The result, 4/5 liters, represents a specific portion of the total soft drink Arjun had, providing a concrete value for the amount he consumed.
Step 2: Soft drink consumed by Naina
Similarly, we calculate the amount of soft drink Naina drank. Since Naina also had 2 liters of soft drink and drank 2/5 of it, the calculation is identical to Arjun's:
(2/5) * 2 liters = 4/5 liters
Therefore, Naina also drank 4/5 liters of soft drink. This repetition of the calculation reinforces the concept of applying the same fraction to the same quantity, leading to the same result. It underscores the symmetry in their consumption and sets the stage for calculating their combined consumption.
Step 3: Total soft drink consumed
To find the total amount of soft drink they both drank, we add the amount Arjun drank to the amount Naina drank:
4/5 liters + 4/5 liters = 8/5 liters
This addition of fractions with the same denominator is a straightforward process, where we simply add the numerators and keep the denominator the same. The result, 8/5 liters, represents the combined consumption of soft drink by Arjun and Naina. We can also express this improper fraction as a mixed number to get a better sense of the quantity.
Step 4: Converting to a mixed number (optional)
The fraction 8/5 can be converted to a mixed number. To do this, we divide 8 by 5. The quotient is 1, and the remainder is 3. Therefore, 8/5 liters is equal to 1 and 3/5 liters.
This conversion provides an alternative representation of the total amount consumed, making it easier to visualize the quantity. 1 and 3/5 liters is more intuitive than 8/5 liters for many people, as it directly states the whole number of liters and the remaining fractional part. This step highlights the flexibility in representing fractions and the importance of choosing the representation that best suits the context.
Conclusion for Part A
In conclusion, Arjun and Naina drank a total of 8/5 liters, or 1 and 3/5 liters, of soft drink. This calculation involved understanding how to multiply a fraction by a whole number and how to add fractions with the same denominator. The result provides a concrete answer to the first part of the problem, quantifying their combined consumption of soft drink.
Part B: Fraction of Total Soft Drink Consumed
Step 1: Total quantity of soft drink
To determine the fraction of the total quantity of soft drink they consumed, we first need to calculate the total amount of soft drink they had initially. Arjun had 2 liters, and Naina had 2 liters, so together they had:
2 liters + 2 liters = 4 liters
This simple addition provides the baseline for calculating the fraction of consumption. The total quantity, 4 liters, serves as the denominator in our fraction, representing the whole amount from which a portion was consumed. This step underscores the importance of identifying the total quantity before calculating fractional parts.
Step 2: Fraction of soft drink consumed
We know they drank a total of 8/5 liters (from Part A), and they initially had 4 liters. To find the fraction of the total soft drink consumed, we divide the amount they drank by the total amount they had:
(8/5 liters) / (4 liters) = 8/5 ÷ 4
This division of a fraction by a whole number involves understanding how to invert and multiply. It's a critical step in expressing the consumed amount as a proportion of the total amount. The next step will clarify the process of simplifying this division.
Step 3: Simplifying the division
To divide 8/5 by 4, we can rewrite 4 as a fraction (4/1) and then multiply by the reciprocal:
(8/5) ÷ (4/1) = (8/5) * (1/4)
Multiplying fractions involves multiplying the numerators and the denominators separately. This step highlights the mechanical process of fraction multiplication and its application in solving division problems.
Step 4: Multiplying the fractions
(8/5) * (1/4) = 8/20
This multiplication gives us the fraction 8/20, which represents the initial fraction of the total soft drink consumed. However, this fraction can be simplified further to its lowest terms.
Step 5: Simplifying the fraction
The fraction 8/20 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
8 ÷ 4 = 2
20 ÷ 4 = 5
Therefore, the simplified fraction is 2/5.
This simplification process is essential for expressing fractions in their most concise form. It reinforces the concept of equivalent fractions and the importance of reducing fractions to their simplest representation.
Conclusion for Part B
Thus, Arjun and Naina drank 2/5 of the total quantity of soft drink. This result represents the proportion of the total soft drink that was consumed, providing a clear and concise answer to the second part of the problem. The process involved dividing the amount consumed by the total amount and simplifying the resulting fraction.
Final Conclusion
In summary, Arjun and Naina drank a total of 8/5 liters (or 1 and 3/5 liters) of soft drink, which represents 2/5 of the total quantity they had. This problem demonstrates the practical application of fractions in everyday scenarios. By understanding how to perform operations with fractions, such as multiplication, addition, and division, we can solve real-world problems involving proportions and quantities. The problem also highlights the importance of simplifying fractions to their lowest terms for clear and concise communication of results. This exercise in mathematical problem-solving reinforces the value of quantitative reasoning in daily life, whether it's sharing a soft drink or managing larger quantities and proportions in other contexts.
