Converting 2.5 + 23/100 + 7/5 To A Fraction A Simple Guide

by Scholario Team 59 views

Hey guys! Today, we're going to tackle a math problem that involves converting a combination of decimals and fractions into a single fraction. Specifically, we'll be working with the expression 2.5 + 23/100 + 7/5. It might seem a bit daunting at first, but don't worry! We'll break it down step by step, making it super easy to understand. Math can be fun, especially when you know how to approach it. So, let’s dive right in and get this sorted!

Understanding the Basics of Fractions and Decimals

Before we jump into solving the problem, let's quickly refresh our understanding of fractions and decimals. Think of a fraction as a way to represent a part of a whole. It's written as two numbers, a numerator (the top number) and a denominator (the bottom number), separated by a line. For instance, in the fraction 1/2, 1 is the numerator, and 2 is the denominator. It means we have one part out of two equal parts. Fractions are essential in many areas, from cooking to engineering, as they allow us to express quantities that aren't whole numbers.

Now, let’s talk about decimals. Decimals are another way of representing numbers that aren't whole. They use a decimal point to separate the whole number part from the fractional part. For example, 2.5 is a decimal where 2 is the whole number, and .5 represents the fractional part. Decimals are incredibly useful in everyday life, especially when dealing with measurements or money. They provide a precise way to express values, making calculations more straightforward in many situations. Understanding both fractions and decimals is crucial because they are often used interchangeably, and being able to convert between them is a valuable skill.

Step-by-Step Conversion Process

Okay, now that we've brushed up on the basics, let's get down to the nitty-gritty of converting 2.5 + 23/100 + 7/5 into a single fraction. Trust me; it’s like following a recipe – each step leads to the delicious final result (which, in this case, is a neatly simplified fraction!).

Step 1: Convert the Decimal to a Fraction

First up, we need to convert the decimal, 2.5, into a fraction. Remember, decimals are just another way of writing fractions, so this is totally doable. To convert 2.5 to a fraction, think of it as 2 and a half. The '.5' represents a half, which we can write as 1/2. So, 2.5 can be written as 2 1/2. But to make it a proper fraction (or, more accurately, an improper fraction), we need to convert the mixed number. Multiply the whole number (2) by the denominator (2) of the fraction and add the numerator (1). This gives us (2 * 2) + 1 = 5. Place this over the original denominator (2), and voilĂ , we have 5/2. So, 2.5 is equivalent to the fraction 5/2. This step is super important because it sets the stage for combining all the terms into a single fraction. Without this conversion, we'd be trying to mix apples and oranges, and nobody wants that in math!

Step 2: Find the Least Common Denominator (LCD)

Next up, we need to find the Least Common Denominator (LCD) for all the fractions. Why? Because to add or subtract fractions, they need to have the same denominator – it's like making sure everyone's playing on the same field. Our fractions are now 5/2 (from the decimal), 23/100, and 7/5. So, we need to find the LCD for 2, 100, and 5. The LCD is the smallest number that all these denominators can divide into evenly. Let’s break it down. The multiples of 2 are 2, 4, 6, and so on. The multiples of 5 are 5, 10, 15, and so on. And the multiples of 100 are, well, 100, 200, and so on. Looking at these, we can see that 100 is the smallest number that 2, 5, and 100 can all divide into without leaving a remainder. So, 100 is our LCD! Finding the LCD is crucial; it's the magic number that allows us to combine our fractions seamlessly. Think of it as the common language that all fractions understand, making the addition process smooth and accurate.

Step 3: Convert Each Fraction to an Equivalent Fraction with the LCD

Now that we've found our LCD (which is 100), we need to convert each fraction into an equivalent fraction with a denominator of 100. This might sound complicated, but it’s just a matter of multiplying the numerator and denominator of each fraction by the same number to get the desired denominator. Let’s start with 5/2. To get the denominator from 2 to 100, we need to multiply by 50 (since 2 * 50 = 100). So, we also multiply the numerator by 50: 5 * 50 = 250. Thus, 5/2 becomes 250/100. Next, we have 23/100. Lucky for us, it already has the denominator we want, so we don’t need to change it. Finally, let's tackle 7/5. To get the denominator from 5 to 100, we multiply by 20 (since 5 * 20 = 100). So, we multiply the numerator by 20 as well: 7 * 20 = 140. Thus, 7/5 becomes 140/100. Now, we have three fractions – 250/100, 23/100, and 140/100 – all with the same denominator. This is a big step because now we can finally add them together. Converting to equivalent fractions is like resizing puzzle pieces to fit the same board; it’s essential for the next step of adding them up!

Step 4: Add the Fractions

Alright, guys, here comes the fun part – adding the fractions! Now that we have 250/100, 23/100, and 140/100, all we need to do is add the numerators together, keeping the denominator the same. So, we add 250 + 23 + 140. Let's do the math: 250 plus 23 is 273, and then adding 140 gives us 413. So, the sum of the numerators is 413. Now, we put this sum over our common denominator, which is 100. This gives us the fraction 413/100. We've successfully added the fractions together! This step is super satisfying because it's where all our previous work comes together. Adding fractions with a common denominator is like adding apples to apples; it's straightforward and gets us closer to our final answer.

Step 5: Simplify the Fraction (If Possible)

Last but not least, we need to check if our fraction, 413/100, can be simplified. Simplifying a fraction means reducing it to its lowest terms, which makes it easier to understand and work with. To do this, we need to find the greatest common divisor (GCD) of the numerator (413) and the denominator (100). The GCD is the largest number that divides both numbers evenly. Let's think about the factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, and 100. Now, we need to see if any of these divide 413 evenly. After checking, we'll find that none of these factors (other than 1) divide 413 without leaving a remainder. This means that 413 and 100 have no common factors other than 1. When a fraction can’t be simplified any further, we say it's in its simplest form. So, 413/100 is already in its simplest form. This step is like putting the final polish on our work; it ensures that our answer is not only correct but also presented in the most concise and clear way possible. And there you have it – we've successfully converted the expression into a single, simplified fraction!

Final Answer and Summary

So, after all that, the expression 2.5 + 23/100 + 7/5 written as a fraction is 413/100. Awesome job, guys! We took a mix of decimals and fractions and turned it into a single, neat fraction. Remember, the key steps were:

  1. Converting the decimal to a fraction.
  2. Finding the Least Common Denominator (LCD).
  3. Converting each fraction to an equivalent fraction with the LCD.
  4. Adding the fractions.
  5. Simplifying the fraction (if possible).

By following these steps, you can tackle any similar problem with confidence. Math is all about breaking things down into manageable steps, and you’ve just nailed it!

Why is This Important?

You might be wondering,