Show a complete solution for each mumber problem. See Example 2 One-fifth of a number plus one-s

Show a complete solution for each mumber problem. See...

Key Concepts

Word Problem Translation

This concept involves converting the language of a real-world problem into a mathematical equation. In the example, phrases like 'one-fifth of a number' and 'one-sixth of a number' are translated into algebraic expressions, which collectively represent the problem statement. This process is fundamental in algebra because it connects everyday scenarios to formal mathematical procedures.

Combining Like Terms with Fractions

When dealing with expressions that involve fractions, it is important to identify and combine like terms. In the context of the problem, both terms contain the same unknown variable multiplied by different fractional coefficients. Combining these terms requires adding them together, which often involves finding a common denominator or understanding how to directly add fractions with the same variable.

Least Common Denominator (LCD) and Clearing Fractions

In equations with fractions, using the least common denominator (LCD) helps simplify the problem by eliminating the fractions. Multiplying both sides of the equation by the LCD converts the fractional equation into an equation with whole numbers, making it easier to solve. This technique is crucial when the equation involves multiple fractional coefficients.

Solving Linear Equations

After forming the equation and combining like terms, the final step is to isolate the variable using inverse operations. Solving linear equations involves performing the same operation on both sides of the equation to maintain equality, thereby finding the value of the unknown variable. This method is central to algebra, as it provides a systematic approach to finding solutions to problems represented by linear equations.

Transcript

00:01 In this problem, we're asked to solve given the following information.

00:04 So we have one -fifth of a number plus one -sixth of that same number is 33.

00:12 And we need to form an equation out of it.

00:15 So i know that a number can be represented by x.

00:19 So that is our mystery number.

00:21 And it looks like we're going to have two of them because i have our mystery number showing up in two different places.

00:28 So let's deal with our first one here.

00:31 So first we have one -fifth of a number.

00:34 So in this case, when i see the word of, that's going to indicate multiplication.

00:41 So i need to have one -fifth multiplied by this number x.

00:46 And i can write that by saying one -fifth x, one -fifth of a number.

00:52 Now we have the word plus that indicates addition.

00:56 So 1 5th x plus 1 6th of the same number.

01:01 So again, i have the word of, and that means multiplication.

01:06 What are we multiplying? well, we're multiplying 1 .6 and our mystery number.

01:12 So 1 .6 multiplied by our mystery number, x, is or equals 33.

01:22 So now all i need to do is combine my like terms, with my x variables and solve for x.

01:31 So let's go ahead and add our 1 5th and 1 6th.

01:36 And in order to add 2 fractions, i need to have a common denominator.

01:40 So it looks like the least common denominator for 5 and 6 is going to be 30.

01:47 So i'm going to go ahead and multiply top and bottom of my 1 5th by 6th, sorry, by 6.

01:56 And i'm going to multiply top and bottom of my one six by five.

02:02 That'll give me common denominators of 30.

02:05 So six times one is six, six times five is 30.

02:10 So six -thirtieth -x plus five times one is five over five times six is 30x.

02:23 And those added together equal 33.

02:28 So now i'm going to go ahead and add my two x variables.

02:33 And i'll start again up here...

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