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To do this, let \(x=\) the repeating fraction, and then we’ll figure out ways to multiply \(x\) by ) just to the right of the decimal point; we get \(10x=4.\underline2525…\).Now we have to line up and subtract the two equations on the left and solve for \(x\); we get \(\displaystyle x=\frac\). Let’s see if it works: Put \(\displaystyle \frac\) in your graphing calculator, and then hit Enter; you should something like \)),“is no more than” (\(\le \)), “is at least” (\(\ge \)), and “is at most” (\(\le \)).
We will see later that this is like a Slope that we’ll learn about in the Coordinate System and Graphing Lines including Inequalities section.
Here’s the math: To get the rate of minutes to photos, we can set up a proportion with the minutes on the top and the photos on the bottom, and then cross multiply.
Now let’s do some problems that use some of the translations above.
We’ll get to more difficult algebra word problems later. Solution: We always have to define a variable, and we can look at what they are asking.
Solution: First we define a variable \(h\), which will be the number of hours that Erica must study (look at what the problem is asking).
We know from above that “at least” can be translated to “\(\ge\)”.The problems here only involve one variable; later we’ll work on some that involve more than one.Doing word problems is almost like learning a new language like Spanish or French; you can basically translate word-for-word from English to Math, and here are some translations: Note that most of these word problems can also be solved with Algebraic Linear Systems, here in the Systems of Linear Equations section.The problem is asking for a number, so let’s make that \(n\).Now let’s try to translate word-for-word, and remember that the “opposite” of a number just means to make it negative if it’s positive or positive if it’s negative.Then \(10-J\) equals the number of pounds of the chocolate candy. This one is a little more difficult since we have to multiply across for the Total row, too, since we want a Don’t worry if you don’t totally get these; as you do more, they’ll get easier.We’ll do more of these when we get to the Systems of Linear Equations and Word Problems topics.Probably the most common is to set up a proportion like we did here earlier. There’s another common way to handle these types of problems, but this way can be a little trickier since the variable in the equation is not what the problem is asking for; we will make the variable a “multiplier” for the ratio.The advantage to this way is we don’t have to use fractions. We can find out how much of ingredients a and b are in solution X by using a ratio multiplier again (one ounce of solution X contains ingredients a and b in a ratio of Let \(x=\) what you need to make on the final.This makes sense, since consecutive means “in a row” and we’re always adding Solution: Let’s first define a variable, and use another table like we did before.Let \(J=\) the number of pounds of jelly candy that is used in the mixture. Let’s put this in a chart again – it’s not too bad.