Comparing Quantities: Orange Juice vs Mango Juice - A Comprehensive Guide

If you have 5/3 litres of orange juice and 13/8 litres of mango juice, which one is in greater quantity and by how much? In this article, we will walk you through a step-by-step process to compare the quantities of orange and mango juice. We will also explore the method of converting fractions to a common denominator and the importance of Least Common Multiple (LCM) in mathematics. So, let’s dive in!

Introduction to the Problem

We have two different fractions representing the quantity of liquid in litres:

5/3 litres of orange juice 13/8 litres of mango juice

To determine which quantity is greater, we need to convert these fractions to a common denominator. This allows us to directly compare their numerators and identify the larger quantity.

Converting Fractions to a Common Denominator

The first step is to find a common denominator for the fractions 5/3 and 13/8. The Least Common Multiple (LCM) of 3 and 8 is 24. Let’s convert each fraction to an equivalent fraction with the denominator 24.

Step 1: Converting 5/3 to a fraction with denominator 24

First, we multiply the numerator and the denominator by 8 (since 3 x 8 24):

5/3  (5 x 8) / (3 x 8)  40/24

Step 2: Converting 13/8 to a fraction with denominator 24

Next, we multiply the numerator and the denominator by 3 (since 8 x 3 24):

13/8  (13 x 3) / (8 x 3)  39/24

Now that we have both fractions with the same denominator, we can directly compare their numerators:

5/3 (or 40/24) litres of orange juice 13/8 (or 39/24) litres of mango juice

Comparison and Conclusion

By comparing the numerators, we can conclude that 40 is greater than 39. Therefore, the quantity of orange juice (40/24 litres) is more than the quantity of mango juice (39/24 litres).

To determine by how much, we subtract the numerators:

40 - 39  1

Thus, the difference in quantity is 1/24 litres. So, the orange juice is greater in quantity by 1/24 litres.

Excel in Quantity Comparison Using LCM and Fractions

Here are the step-by-step methods to solve such problems:

Method 1: Direct Fraction Conversion

Convert 5/3 to 40/24 Convert 13/8 to 39/24 Subtract: 40/24 - 39/24 1/24 litres

As a result, orange juice is more than mango juice by 1/24 litres.

Method 2: Decimal Conversion

5/3 1.6666667 litres (orange juice) 13/8 1.625 litres (mango juice) Difference: 1.6666667 - 1.625 0.0416666 litres (which is 1/24 litres)

Again, the orange juice is more in quantity by 1/24 litres.

Additional Insight

Suppose you have ( x ) litres of orange juice and ( y ) litres of mango juice. To know which one is in greater quantity, you need to compare ( x ) and ( y ). Given ( x frac{5}{3} ) and ( y frac{13}{8} ), we can proceed as follows:

x - y  frac{5}{3} - frac{13}{8}  frac{40 - 39}{24}  frac{1}{24}

Clearly, (frac{1}{24}) is positive, indicating that the orange juice is more in quantity by (frac{1}{24}) litres.

Conclusion

By carefully converting fractions to a common denominator and using the Least Common Multiple, you can accurately compare the quantities of different liquids or any other items represented as fractions. Understanding these basic mathematical concepts will be invaluable in solving a wide range of problems, from everyday comparisons to complex calculations in science, engineering, and other fields.

Key Takeaways

Convert fractions to a common denominator using LCM Subtract the numerators to find the difference in quantities Use decimal conversion for a more intuitive understanding

Remember, the difference in quantity can be expressed in fractions or decimals, and both represent the same relationship. Always verify your results to ensure accuracy. Happy comparing!