It's a common puzzle that pops up in math classes, and sometimes even in everyday problem-solving: you're given a fraction of a number, and you know what that fraction equals, but you need to find the original whole number. Let's say we're told that a certain fraction of a number is equal to 120. How do we go about finding that original number?
Think of it like this: if you have a pizza, and you're told that two-thirds of the pizza weighs 120 grams, you'd naturally want to know how much the whole pizza weighs, right? The same logic applies here.
When we're dealing with fractions, we often use multiplication and division. If a fraction of a number equals a certain value, it means we're essentially multiplying that fraction by the unknown number to get the result. So, if we represent our unknown number as 'x', and the fraction is, say, , then the equation would look something like this:
(2/3) * x = 120
Now, how do we isolate 'x' to find out what it is? We need to do the opposite of multiplying by 2/3. The opposite of multiplying by a fraction is dividing by that fraction, or, more commonly, multiplying by its reciprocal. The reciprocal of 2/3 is 3/2.
So, we'd multiply both sides of our equation by 3/2:
x = 120 * (3/2)
Let's break that down. We can think of 120 as 120/1. So, we're multiplying (120/1) by (3/2). When we multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together:
x = (120 * 3) / (1 * 2)
x = 360 / 2
x = 180
So, in this example, the original number would be 180. If two-thirds of 180 is 120, then the whole number is indeed 180.
Another way to look at it, as seen in some of the examples, is when the fraction is different. For instance, if we're told that a number's is equal to 120. Here, the fraction is . So, we'd set up the equation:
(1/5) * x = 120
To find 'x', we multiply both sides by the reciprocal of 1/5, which is 5/1 (or just 5):
x = 120 * 5
x = 600
It's a straightforward process once you understand the relationship between fractions and the whole. It's all about reversing the operation. If a part of something is given, you use division (or multiplication by the reciprocal) to find the whole. It's a fundamental concept in arithmetic that helps us solve a variety of problems, from simple math exercises to more complex real-world scenarios.
