Question

How do you evaluate \[3\dfrac{5}{9} - 2\dfrac{2}{5}\] ?

Answer 1

Hint: Mixed fractions are the combination of a whole number and a proper fraction. To evaluate the given mixed number, we need to convert the given mixed number to an improper fraction i.e., multiply the whole number by the denominator of a fraction. Add that to the numerator and then write the obtained terms on top of the denominator, hence you get an improper fraction then a mixed fraction.

Complete step-by-step answer:
 Given,
 \[3\dfrac{5}{9} - 2\dfrac{2}{5}\] , in which we need to evaluate the given numbers.
Hence, first we need to convert the given mixed number to an improper fraction. As, a mixed number is an addition of its whole and fractional parts i.e., we need to add both the numbers separately.
Let, us consider the first term: \[3\dfrac{5}{9}\]
As, mentioned in the above statement; add 3 and \[\dfrac{5}{9}\] as:
 \[ = 3 + \dfrac{5}{9}\]
To write 3 as a fraction with a common denominator, multiply by \[\dfrac{9}{9}\] as:
 \[ = 3 \cdot \dfrac{9}{9} + \dfrac{5}{9}\]
Combine 3 and \[\dfrac{9}{9}\] we get:
 \[ = \dfrac{{3 \cdot 9}}{9} + \dfrac{5}{9}\]
Combine the numerators over the common denominator as:
 \[ = \dfrac{{3 \cdot 9 + 5}}{9}\]
Simplify the numerator terms i.e., multiply 3 and 9 as:
 \[ = \dfrac{{27 + 5}}{9}\]
Our next step is to add the numerator terms, 27 and 5, giving us the desired improper fraction:
 \[ = \dfrac{{32}}{9}\]
Now, let us consider the second term: \[2\dfrac{2}{5}\]
As, mentioned in the above statement; add 2 and \[\dfrac{2}{5}\] as:
 \[ = 2 + \dfrac{2}{5}\]
To write 2 as a fraction with a common denominator, multiply by \[\dfrac{5}{5}\] as:
 \[ = 2 \cdot \dfrac{5}{5} + \dfrac{2}{5}\]
Combine 2 and \[\dfrac{5}{5}\] we get:
 \[ = \dfrac{{2 \cdot 5}}{5} + \dfrac{2}{5}\]
Combine the numerators over the common denominator as:
 \[ = \dfrac{{2 \cdot 5 + 2}}{5}\]
Simplify the numerator terms i.e., multiply 2 and 5 as:
 \[ = \dfrac{{10 + 2}}{5}\]
Our next step is to add the numerator terms, 10 and 2, giving us the desired improper fraction:
 \[ = \dfrac{{12}}{5}\]
Hence, we got the improper fraction of the given numbers as:
 \[3\dfrac{5}{9} = \dfrac{{32}}{9}\] and \[2\dfrac{2}{5} = \dfrac{{12}}{5}\]
Next, put each fraction over a common denominator as we have 5 and 9 as the common denominator for the:
First term:
 \[\dfrac{{32}}{9} = \dfrac{5}{5} \times \dfrac{{32}}{9}\]
To multiply combine the numerators over the common denominator as:
 \[ = \dfrac{{5 \times 32}}{{5 \times 9}}\]
 \[ = \dfrac{{160}}{{45}}\] ……………… 1
And for, Second term:
 \[\dfrac{{12}}{5} = \dfrac{9}{9} \times \dfrac{{12}}{5}\]
To multiply combine the numerators over the common denominator as:
 \[ = \dfrac{{9 \times 12}}{{9 \times 5}}\]
 \[ = \dfrac{{108}}{{45}}\] ………………… 2
Then, rewrite the expression and subtract the numerators over the common denominator from equation 1 and 2 as:
 \[3\dfrac{5}{9} - 2\dfrac{2}{5} = \dfrac{{160}}{{45}} - \dfrac{{108}}{{45}}\]
Combine the numerators over the common denominator as:
 \[ = \dfrac{{160 - 108}}{{45}}\]
Hence, we get the exact form as:
 \[ = \dfrac{{52}}{{45}}\]
Now, convert the obtained improper fraction into a mixed number:
 \[\dfrac{{52}}{{45}}\]
Divide the numerator by the denominator term as we have 45 as the denominator term, hence:
 \[ = \dfrac{{45 + 7}}{{45}}\]
Separate the terms to get the mixed fraction i.e.,
 \[ = \dfrac{{45}}{{45}} + \dfrac{7}{{45}}\]
As, simplifying the terms we get:
 \[ = 1 + \dfrac{7}{{45}}\] \[ \Rightarrow 1\dfrac{7}{{45}}\]
Hence,
 \[3\dfrac{5}{9} - 2\dfrac{2}{5} = 1\dfrac{7}{{45}}\]
So, the correct answer is “$1\dfrac{7}{{45}}$”.

Note: To evaluate the given a mixed number, we need to convert it to an improper one, we must transfer the whole number into the numerator of the fraction. In improper fraction, the numerator is greater than the denominator, hence note that improper fraction is always greater than one. A proper fraction has a numerator smaller than the denominator but an improper fraction has a denominator smaller than the numerator.