to view answer.To see more practice with fractions, follow the links:
Adding and Subtracting Fractions
Multiplying and Dividing Fractions
The fractions $$\frac{16}{20}$$, $${\Large\frac{24}{30}}$$, $${\Large\frac{48}{60}}$$, and $${\Large\frac{400}{500}}$$ all represent the same value. This is because the ratio between the numerator and the denominator is the same for each. The proportion between the top and the bottom of each is the same. Each numerator is $$\frac{4}{5}$$ of the denominator. Thus, each of these fractions represent the same part of the whole, or $${\Large\frac{4}{5}}$$ of the whole. Therefore, they can each be simplified to $${\Large\frac{4}{5}}$$.
For any fraction, if you multiply the numerator and denominator by the same number, the fraction still has the same value. This is because the proportion between the top and bottom remains the same. You are actually just multiplying the fraction by 1. For example:
$${\Large\frac{13}{20}}={\Large\frac{5}{5}} \cdot{\Large \frac{13}{20}} ={\Large\frac{65}{100}}$$
The part-to-whole ratio of 13 to 20 is the same as the part-to-whole ratio of 65 to 100. We can also think of this as scaling up both the top and the bottom of the fraction by the same multiple. This scaling does not change the proportion, and so the fraction still has the same value.
When you are asked to simplify a fraction, you want to find the fraction that has the lowest possible terms for the numerator and denominator. The simplified version of the fraction needs to have the same value as the original fraction. Thus, we scale down the numerator and denominator values by dividing each by the same number (or multiplying each by the same fraction) until there are no more numbers to divide out.
Simplify:
$$
{\Large{{210} \over {350}}}
$$
1. Note that the numerator and denominator both end in zero, so they are divisible by 10. So, divide each by 10:
$$
{\Large{{210 \div 10} \over {350 \div 10}} = {{21} \over {35}}}
$$
Another look at why this is true: We know that multiplying by 1 does nothing to a number. In fact, 1 is the multiplicative identity element – it preserves the identity of the number that it is being multiplied by. Also, note that dividing by 10 is the same as taking $$\frac{1}{10}$$ of a number, or, in other words, multiplying by $$\frac{1}{10}$$. So, if you divide both the top and the bottom of a fraction by the 10, that is same as multiplying both the top and the bottom by $$\frac{1}{10}$$. So, what we have here is: $$\frac{210 \div 10}{350 \div 10}=\frac{210 \cdot \frac{1}{10}}{350 \cdot \frac{1}{10}}=\frac{21}{35}$$ So, dividing the numerator and denominator by 10 is the same as multiplying the fraction by 1. This does nothing to the value of the fraction. All we’ve done is change its form (and made it simpler). |
2. Now, notice that both 21 and 35 are divisible by 7. So, divide each by 7 (or multiply each by $${\Large\frac{1}{7}}$$):
$${\Large\frac{21 \div 7}{35 \div 7}}={\Large\frac{21 \cdot \frac{1}{7}}{35 \cdot \frac{1}{7}}}={\Large\frac{3}{5}}$$
3. Then, $${\Large\frac{3}{5}}$$ is the final, simplified fraction because there is nothing that divides both 3 and 5.
The box below has some divisibility rules to help you determine divisors of large numbers.
Finding common divisors: There are some simple checks to see if numbers are divisible by certain other numbers: Divisibility Rules: 1: Every number is divisible by 1. However, this does not help us when simplifying fractions because dividing by 1, like multiplying by 1, does nothing to the number, and so does nothing to help change its form. 2: If the last digit of a number is even, then the number is divisible by 2. 3: If the sum of the digits of a number is divisible by 3, then the number, itself, is divisible by 3. For example: 24,732,120 is divisible by 3 since: 2+4+7+3+2+1+2+0=21 and 21 is divisible by 3. 4: If the last 2 digits of a number are a number that is divisible by 4, then the whole number is divisible by 4. For example: 2,398,748 is divisible by 4 because 4 and 8 are the last two digits, and 48 is divisible by 4. 5: If the number ends in 5 or 0, it is divisible by 5. For example: 248,486,575 and 32,987,460 are divisible by 5. 6: If the number is divisible by 3 (the sum of its digits is divisible by 3) and it is also divisible by 2 (the last digit is even), then the number is divisible by 6. For example: 24,732,120 is divisible by 6 because, as shown above, it is divisible by 3, and also it ends in 0, which is even, so it is divisible by 2. 7: If the sum of the digits of a number is divisible by 9, then the number, itself, is divisible by 9. For example: 2,875,653 is divisible by 9 since: 2+8+7+5+6+5+3=36, and 36 is divisible by 9. 8: If the number ends in 0, it is divisible by 10. For example: 239,847,590 is divisible by 10. Okay, but what if the numbers you are looking at is not divisible by any of the above numbers? Does that mean that it is prime (which means divisible by only 1 and itself)? Absolutely not! There are plenty of numbers that it could be divisible by. The best way to go about looking for divisors other than those above is to start dividing by prime numbers. This will ensure that you are not double-checking things that you have already checked. For instance, if you check to see if the number is divisible by 15, this would be a waste of time, if you have already checked 5 (since 15 is divisible by 5). So, start with: 7, 11, 13, 17, 23, 29… You probably won’t have to go further than that. |
Now, we look at an example of a fraction that includes variables.
Simplify:
$${\Large\frac{105x^2y}{245xy^3}}$$
1. First, let’s simplify the numerical part of the fraction. Notice that both 105 and 245 end in 5, so they must both be divisible by 5. So, we start by dividing the numerator and denominator by 5:
$${\Large\frac{105x^2y \div 5}{245xy^3 \div 5}} = {\Large\frac{\frac{105x^2y}{5}}{\frac{245xy^3}{5}}} ={\Large \frac{21x^2y}{49xy^3}}$$
Next, notice that 21 and 49 are both multiples of 7. So, divide each by 7:
$${\Large\frac{21x^2y \div 7}{49xy^3 \div 7} }= {\Large\frac{\frac{21x^2y}{7}}{\frac{49xy^3}{7}}} = {\Large\frac{3x^2y}{7xy^2}}$$
Finally, both 3 and 7 are prime (they have no divisors except 1 and themselves), so we can divide nothing else out of both of them, and can therefore simplify the numerical part of the fraction no further.
2. Next, we need to simplify the part of the fraction involving variables. Right now, our problem looks like:
$${\Large\frac{3x^2y}{7xy^3}}$$
In the numerator, we have $$x^2$$, and we know that $$x^2 = x \cdot x$$. Similarly, in the denominator, we have $$y^3$$, and we know that $$y^3 = y \cdot y \cdot y$$\
Rewriting the fraction, we have:
$${\Large\frac{3 \cdot x \cdot x \cdot y}{7 \cdot x \cdot y \cdot y \cdot y}}$$
This can be rewritten as:
$${\Large\frac{3 \cdot x}{7 \cdot y \cdot y}} \cdot {\Large\frac{x}{x}} \cdot {\Large\frac{y}{y}}$$
(See multiplying and dividing fractions if you are not sure why this is true).
Then, since $${\Large\frac{x}{x}} = 1$$ and, similarly, $${\Large\frac{y}{y}} =1$$, we can write:
$${\Large\frac{3 \cdot x}{7 \cdot y \cdot y}} \cdot {\Large\frac{x}{x}} \cdot {\Large\frac{y}{y}} = {\Large\frac{3 \cdot x}{7 \cdot y \cdot y}} \cdot 1 \cdot 1 ={\Large \frac{3x}{7y^2}} $$
This is the reason that we can “cancel” multiplied terms from the numerator and denominator. When “cancelling”, we are just dividing out 1’s.
3. Thus, our final, simplified fraction is: $${\Large\frac{3x}{7y^2}}$$
Some practice problems to check your skills:
1. Simplify the fraction: $$
{\Large{{108} \over {180}}}
$$
Think about it for a moment and then access this link to view answer.
2. Simplify the fraction: $$
{\Large{{196,000} \over {252,000}}}
$$
Think about it for a moment and then access this link to view answer.
3. Simplify: $$
{\Large{{72x^3 y} \over {78x^3 }}}
$$
Think about it for a moment and then access this link to view answer.
4. Simplify: $$
{\Large{{95a^5 b^2 } \over {115a^3 b}}}
$$
Think about it for a moment and then access this link to view answer.
5. Simplify: $$
{\Large{{900xyz} \over {1020x^7 }}}
$$
Think about it for a moment and then access this link to view answer.