HOW TO SOLVE FRACTION PROBLEMS

Fraction

A fraction is part of a whole. A piece from a thing. Anything piece from an object.

This is one basic fundamental of mathematics that should not be ignored. It is very useful and usable in many fields. From engineering, business, and more on various fields involves mathematical computations. Everything we see is the composition of fractions. Let's talk baking as an example. In baking, we need fractions to apply the proportion of its some other ingredients and the time to make it perfect. We eat, and we barely notice how the food was prepared. Its proportion is personally fractionally operated according to its right proportionality. Infrastructures, say building constructions as a usual example. From the mixing of cement, the number of workers works per week, time allotments, and a lot more contributing to finish one construction, all of them were the product of fractional operations. In short, all man-made project designs around us were composed of basic to complex fractions. Can we imagine if we do not pay attention to fractions? Can we think what happens if fraction has least focused on our daily lives? The importance of fraction is usually underrated because we pay attention to higher mathematics such as Algebra, Calculus, Geometry, etc. but all of them grew fractionally.

There are only four basic operations to deal with this. Addition, Subtraction, Multiplication, and Division. Some operations that involve the complexity of operations still use the four basic operations. Fractions vary by type. A proper fraction is a type of fraction that the numerator is less than its denominator and for the improper fraction its the inverse of the improper fraction where the denominator is less than its denominator. There also exist mixed numbers or also called mixed fractions as an alternate term, where there are one whole number and a proper fraction. A mixed number is composed of an integer and a proper fraction. It is an improper fraction converted into a mixed number. To illustrate, see illustrations below;

Illustrations of the following:

I.) proper fraction - this fraction has the numerator is always less than its denominator.

$\frac{1}{2},\frac{3}{4},\frac{7}{10},\frac{15}{16}, etc.$ , notice that the numerator < the denominator.

II.) improper fraction- this fraction has the numerator is always greater than its denominator.

$\frac{3}{2},\frac{4}{3},\frac{8}{5},\frac{11}{7}, etc.$ , notice that the numerator > the denominator.

III.) mixed numbers- this is the combination of a whole number (integer) and a proper fraction.

$1\frac{1}{2},2\frac{2}{3},4\frac{1}{5}, 6\frac{3}{7}, etc.$ , notice that there are an integer and a proper fraction

Additional Photo was provided below to distinguish between those mentioned fraction types.

Operations on Fractions:

Usually, the majority don't like finding the LCD. So this might help as an alternate way of solving fraction operations. This will teach how the operation of fraction works. We have to follow the position of every letter provided then assigned numbers to correspond to it.

The general form for addition and subtraction of dissimilar fraction found below: Note that dissimilar fractions are those fractions with no denominators equal.

$\frac{{\color{Red} a}}{{\color{Blue} b}}\pm \frac{{\color{DarkGreen} c}}{{\color{Magenta} d}}=\frac{{\color{Red} a}{\color{Magenta} d}\pm {\color{Blue} b}{\color{DarkGreen} c}}{{\color{Blue} b}{\color{Magenta} d}}$ , this is good when you don't like to find the least common denominator (LCD). Thus this is made for that intention.

For Addition (similar fraction): Note that similar fractions are those fractions whose denominators are equal.

$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$

Subtraction (similar fraction):

$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}$

Multiplication:

$\frac{a}{b}*\frac{c}{d}=\frac{a*c}{b*d}$

Division:

$\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}x\frac{d}{c}=\frac{ad}{bc}$ , just flip the second fraction upside down, and change the division into multiplication as their operator.

Steps on how to do it:

Perform the indicated operations:

$1.)\frac{1}{2}+\frac{3}{4}=$ , so if we wanna solve this fraction expressions for simplifying we can apply what we have learned from the previous discussions. Let's assign each number to the formula; See below;

$Let :a=1, b=2,c=3,d=4$

So let's substitute their corresponding values to its formula found by

$\frac{{\color{Red} a}}{{\color{Blue} b}}\pm \frac{{\color{DarkGreen} c}}{{\color{Magenta} d}}=\frac{{\color{Red} a}{\color{Magenta} d}\pm {\color{Blue} b}{\color{DarkGreen} c}}{{\color{Blue} b}{\color{Magenta} d}}$

Now,

$\frac{1}{2}+\frac{3}{4}=\frac{1(4)+2(3)}{2(4)}=$

$=\frac{4+6}{8}=\frac{10}{8}$

$=\frac{{\color{DarkGreen} 2}*5}{{\color{DarkGreen} 2}*4}$

$=\frac{5}{4}$

$=2\frac{1}{2}$ , this the final answer. But of course, we can have the easiest way of implying fractions by finding the Least Common Denominator (LCD). Finding the LCD is by thinking of the least number than can be divisible by all denominators. That number (LCD) is 4 because 4 is divisible by 2 as the first denominator and the second denominator 4. See the simple operation below.

$4(\frac{1}{2}={\color{Red} 2})$ and $4(\frac{3}{4}={\color{Red} {\color{Blue} 1}})$ , this is to get the number to be multiplied for the first and second numerators of this fraction operation respectively.

$\frac{1({\color{Red} 2})}{2}+\frac{3({\color{Blue} 1})}{4}=\frac{1({\color{Red} 2})+3({\color{Blue} 1})}{4}$ , look at how they were applied to the operation. They're in colors.

$\frac{1({\color{Red} 2})}{2}+\frac{3({\color{Blue} 1})}{4}=\frac{1({\color{Red} 2})+3({\color{Blue} 1})}{4}=\frac{{\color{Red} 2}+{\color{Blue} 3}}{4}=\frac{5}{4}$ , operation continuation.

$\frac{1({\color{Red} 2})}{2}+\frac{3({\color{Blue} 1})}{4}=\frac{1({\color{Red} 2})+3({\color{Blue} 1})}{4}=\frac{{\color{Red} 2}+{\color{Blue} 3}}{4}=\frac{5}{4}=2\frac{1}{2}$ , arrived at the final answer.

Let's try more.

$2.)\frac{3}{4}-\frac{1}{5}$

So by using the formula:

Let

${\color{Red} a}=3,{\color{Blue} b}=4,{\color{DarkGreen} c}=1,{\color{Magenta} d}=5$

So we have,

$\frac{3}{4}-\frac{1}{5}=\frac{3(5)-4(1)}{4(5)}=\frac{15-4}{20}=\frac{11}{20}$ , so this is the final answer.

$3.) \frac{4}{5}+\frac{1}{5}=\frac{5}{5}=1$ , this is similar fractions so just copy the common denominator and simply perform the indicated operation fo the numerators.

$4.) \frac{2}{3}*\frac{6}{5}=\frac{2}{{\color{DarkGreen} 3}}(\frac{2*{\color{DarkGreen} 3}}{5})=\frac{4}{5}$ , this is the multiplication of fractions. There's no problem if they are similar or dissimilar fractions. The process is simple, multiply the numerator of the first fraction to the numerator of the second fraction, and denominator to denominator.

$5.) \frac{2}{5}\div \frac{4}{3}=\frac{2}{5}*\frac{3}{4}=\frac{6}{20}=\frac{3}{10}$ , this is a division of fractions. The process is simply flipped upside down the second fraction, and change its operator from division to multiplication. Now follow the rule for multiplication of fractions.

$6.)\frac{1}{2}+\frac{2}{3}+\frac{1}{6}$ , so this is a bit complex of the addition of fractions. If you are good at finding the Least Common Denominator (LCD), this is easy, we found it as "6". Let's try to simplify this by using the LCD.

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{(1*{\color{Red} 3})+(2*{\color{Red} 2})+(1*{\color{Red} 1})}{6}$ , the color in red is the quotient of the LCD "6". So we divide the first denominator "2" from the first fraction to the LCD "6", and gave us "3" and now the multiplier of the first's fractions' numerator "1". We do the pattern on the second and third fractions in the series.

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{(1*{\color{Red} 3})+(2*{\color{Red} 2})+(1*{\color{Red} 1})}{6}=\frac{{\color{Red} 3}+{\color{Red} 4}+{\color{Red} 1}}{6}=\frac{{\color{Red} 8}}{6}=\frac{{\color{DarkGreen} 2}*4}{{\color{DarkGreen} 2}*3}$ , to simplify, we factor both numerators and denominators. Now we canceled out "2" as their common factor.

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{(1*{\color{Red} 3})+(2*{\color{Red} 2})+(1*{\color{Red} 1})}{6}=\frac{{\color{Red} 3}+{\color{Red} 4}+{\color{Red} 1}}{6}=\frac{{\color{Red} 8}}{6}=\frac{{\color{DarkGreen} 2}*4}{{\color{DarkGreen} 2}*3}=\frac{4}{3}$ , so we have the result 4/3 but we can simplify this into a mixed number.

So let's try using our rule

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}$ , so to make it easier, let's work on the first two factions first.

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{(1*3)+(2*2)}{2*3}+\frac{1}{6}$ , this is how it looks like.

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{(3)+(4)}{6}+\frac{1}{6}$ , simplifying numerators and denominators by multiplying all of them in accordance with their respective position.

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{7}{6}+\frac{1}{6}$ , after doing such, notice that we have a similar fraction to simplify now. So we can simply add numerators and copy denominators which is common.

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{7}{6}+\frac{1}{6}=\frac{7+1}{6}$ , this is the result.

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{7}{6}+\frac{1}{6}=\frac{7+1}{6}=\frac{8}{6}$ , the 8/6 is an answer need to be simply or reduce to its lowest term. So,

$\frac{1}{2}+\frac{2}{3}+\frac{1}{6}=\frac{7}{6}+\frac{1}{6}=\frac{7+1}{6}=\frac{8}{6}=\frac{{\color{DarkGreen} 2}*4}{{\color{DarkGreen} 2}*3}=\frac{4}{3}=1\frac{1}{3}$ , this is the final answer. Notice that the answer is the same as having an LCD to simplify. But as we have said, there are students that don't like to find it. This is an alternative way but we encourage finding LCD is easier. Just need a little constant practice.

Let's try multiplication of fractions with three fractions in the series.

$7.)\frac{2}{5}*\frac{1}{4}*\frac{3}{2}$ , this is our example as a basic simple fraction. As the rule says, we need to multiply directly all numerators to each other and do the same to our denominators. But one advice is that we need to cancel out earlier the common factor found from numerators and denominators to make our operation easier. Now let's do it.

$\frac{2}{5}*\frac{1}{4}*\frac{3}{2}=\frac{{\color{DarkGreen} 2}}{5}*\frac{1}{4}*\frac{3}{{\color{DarkGreen} 2}}$ , "2" are the common factor. We colored it in green for an easier spot.

$\frac{2}{5}*\frac{1}{4}*\frac{3}{2}=\frac{{\color{DarkGreen} 2}}{5}*\frac{1}{4}*\frac{3}{{\color{DarkGreen} 2}}=\frac{1*3}{5*4}$ , so what we have now is 1 and 3 for the numerator, and 5 and 4 for our denominator. Finally,

$\frac{2}{5}*\frac{1}{4}*\frac{3}{2}=\frac{{\color{DarkGreen} 2}}{5}*\frac{1}{4}*\frac{3}{{\color{DarkGreen} 2}}=\frac{1*3}{5*4}=\frac{3}{20}$ , so 3/20 is our final answer.

What if we are given a series of divisions? Let's try it.

$8.)\frac{2}{3}\div \frac{1}{2}\div \frac{4}{5}$ , so this has to be solved the way we knew about division simplification. Flipping the second up to the third fraction upside down in the series is the key. See below.

$\frac{2}{3}\div \frac{1}{2}\div \frac{4}{5}=\frac{2}{3}* \frac{2}{1}* \frac{5}{4}$ , this is now the new looks of the series.

$\frac{2}{3}\div \frac{1}{2}\div \frac{4}{5}=\frac{{\color{DarkGreen} 2}}{3}* \frac{{\color{DarkGreen} 2}}{1}* \frac{5}{{\color{DarkGreen} 4}}$ , the highlighted in green is the factors that need to cancel out. Finally,

$\frac{2}{3}\div \frac{1}{2}\div \frac{4}{5}=\frac{{\color{DarkGreen} 2}}{3}* \frac{{\color{DarkGreen} 2}}{1}* \frac{5}{{\color{DarkGreen} 4}}=\frac{5}{3}$ , or

$\frac{2}{3}\div \frac{1}{2}\div \frac{4}{5}=\frac{{\color{DarkGreen} 2}}{3}* \frac{{\color{DarkGreen} 2}}{1}* \frac{5}{{\color{DarkGreen} 4}}=\frac{5}{3}=1\frac{2}{3}$ , final answer.

Example Problems with Solution:

Solve the following fractions. The (*) symbol means multiply:

Click to download this copy here!

$1.)\frac{1}{2}+\frac{3}{4}=\frac{1*4+2*3}{2*4}=\frac{4+6}{8}=\frac{5}{4}={\color{Red} 1\frac{1}{4}}$ , so in this fraction operation, the given fractions are not similar, so it means we need to use the general form of adding it without getting the least common denominator.

$2.) \frac{3}{5}-\frac{1}{2}=\frac{3*2-5*1}{5*2}=\frac{6-5}{10}={\color{Red} \frac{1}{10}}$ , in the problem found above, the operation is addition. So let's try subtraction. And this is the result.

$3.)\frac{2}{3}*\frac{6}{4}=\frac{2*6}{3*4}=\frac{12}{12}={\color{Red} 1}$ , so we used the pattern for multiplication of fraction here.

$4.)\frac{4}{5}\div \frac{8}{10}=\frac{4}{5}*\frac{10}{8}=\frac{40}{40}={\color{Red} 1}$ , This is how we do the division. The second fraction from left to right order was flipped upside down.

$5.)\frac{3}{7}+\frac{2}{3}=\frac{3*3+7*2}{7*3}=\frac{9+14}{21}=\frac{23}{21}={\color{Red} 1\frac{2}{21}}$ , addition of fraction again with general form operation.

$6.)\frac{2}{3}-\frac{1}{4}=\frac{4*2-3*1}{3*4}=\frac{8-3}{12}={\color{Red} \frac{5}{12}}$ , Subtraction of the dissimilar fraction using the general formula.

$7.) \frac{3}{5}x\frac{2}{3}=\frac{3*2}{5*3}={\color{Red} \frac{6}{15}}$ , multiplication of fractions.

$8.) \frac{4}{7}\div \frac{5}{3}=\frac{4}{7}x\frac{3}{5}=\frac{4*3}{7*5}={\color{Red} \frac{12}{35}}$ , division of fraction using the formula given above for division. See how the second fraction flipped upside down?

$9.)\frac{5}{6}+\frac{3}{6}=\frac{5+3}{6}=\frac{8}{6}=\frac{4}{3}={\color{Red} 1\frac{1}{3}}$ , addition of dissimilar fraction.

$10.)\frac{3}{4}-\frac{1}{4}=\frac{3-1}{4}=\frac{2}{4}=\frac{1*{\color{DarkGreen} 2}}{2*{\color{DarkGreen} 2}}={\color{Red} \frac{1}{2}}$ , subtraction of dissimilar fraction.

Problems with Partial Solutions

From the given fractions below, perform the indicated operations:

$1.) \frac{3}{4}+\frac{1}{2}=$

$=\frac{3(2)+1(4)}{4(2)}$

$=\frac{6+4}{8}$

$=$ Click to write your answer

$2.)\frac{2}{3}-\frac{3}{5}=$

$=\frac{2(5)-3(3)}{3(5)}$

$=$ Click to write your answer

$3.) \frac{3}{7}-\frac{1}{7}=$ , similar fraction so it's easy

$=$ Click to write your answer

$4.)\frac{3}{4}*\frac{2}{6}=\frac{3*2}{2*2}*\frac{1*2}{2*3}$

$=$ Click to write your answer

$5.)\frac{5}{6}\div \frac{2}{3}=\frac{5}{6}* \frac{3}{2}$

$=$ Click to write your answer

$6.) \frac{3}{2}+\frac{2}{3}+\frac{1}{4}$

$\frac{3}{2}+\frac{2}{3}+\frac{1}{4}=\frac{(3*{\color{Red} 6})+(2*{\color{Red} 4})+(1*{\color{Red} 3})}{12}$

$=$ Click to write your answer

$7.)\frac{2}{5}*\frac{3}{4}*\frac{5}{6}$

$=\frac{2}{5}*\frac{3}{4}*\frac{5}{2*3}$

$=$ Click to write your answer

$8.)\frac{1}{3}\div \frac{2}{3}\div \frac{3}{4}$

$=\frac{1}{3}* \frac{3}{2}* \frac{4}{3}$

$=$ Click to write your answer

$9.)\frac{3}{4}*\frac{4}{5}*\frac{2}{3}$

$=$ Click to write your answer

$10.)\frac{3}{4}\div \frac{4}{5}\div \frac{2}{3}$

$=$ Click to write your answer

Simple Quiz:

Show your answer in the blank provided.

$1.)\frac{2}{5}+\frac{1}{2}=\frac{2*2+5*1}{5*2}=\frac{4+5}{10}= {\color{Red} ?}$ ________

$2.)\frac{3}{5}-\frac{2}{5}={\color{Red} ?}$ ________

$3.)\frac{5}{7}x\frac{7}{10}= {\color{Red} ?}$ ________

$4.)\frac{7}{9}\div \frac{14}{18}= {\color{Red} ?}$ ________

$5.)\frac{1}{3}+ \frac{3}{5}= {\color{Red} ?}$ ________

$6.)\frac{1}{3}- \frac{3}{5}= {\color{Red} ?}$ ________

$7.)\frac{4}{5}x \frac{10}{4}= {\color{Red} ?}$ ________

$8.)\frac{4}{5}\div \frac{4}{5}= {\color{Red} ?}$ ________

$9.)\frac{4}{5}+ \frac{4}{5}= {\color{Red} ?}$ ________

$10.)\frac{3}{7}- \frac{1}{3}= {\color{Red} ?}$ ________

click for the answers

Click for Downloadable PDF worksheets with the answer key!!!REFERENCES

Related References: Proper Fractions, Improper Fractions, Operations on Fractions

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