Questions & Answers
Question
AnswersFill in the missing fraction $\dfrac{{\boxed{}}}{{\boxed{}}} + \dfrac{5}{{12}} = \dfrac{{12}}{{27}}$
Answer
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Hint: To find the missing fraction let us assume the missing fraction to be ‘x’.
So, to find the value of x, we will solve the given equation of fraction with the given mathematical operations in the equation.
Complete step-by-step answer:
Now, according to the equation given in the question, to find the missing fraction, let the missing fraction be x, then putting ‘x’ in the place of the missing fraction, the equation will change to:
$x + \dfrac{5}{{12}} = \dfrac{{12}}{{27}}$.
Now, on the LHS, x is being added to a fraction.
So, to separate x and solve for x, we have to subtract $\dfrac{5}{{12}}$ from both sides of the equation, so that we are left with just x on the LHS, such that it will be equal to the term on the RHS.
The other way to understand this is, we can simply take $\dfrac{5}{{12}}$to the other side, such that the + sign before $\dfrac{5}{{12}}$will change to a minus sign, when it is taken to the right hand side. So, the equation will now become:
$x = \dfrac{{12}}{{27}} - \dfrac{5}{{12}}$
So now we will subtract $\dfrac{5}{{12}}$ from $\dfrac{{12}}{{27}}$ . Now to do the subtraction on the RHS,
We have to first find out the LCM of 12 and 27, for which we need to find the prime factors of 12 and 27.
So prime factorization of 12 and 27 will give us the factors as:
$
2\left| \!{\underline {\,
{12} \,}} \right. \\
2\left| \!{\underline {\,
6 \,}} \right. \\
3\left| \!{\underline {\,
3 \,}} \right. \\
1\left| \!{\underline {\,
1 \,}} \right. \\
$ $
3\left| \!{\underline {\,
{27} \,}} \right. \\
3\left| \!{\underline {\,
9 \,}} \right. \\
3\left| \!{\underline {\,
3 \,}} \right. \\
1\left| \!{\underline {\,
1 \,}} \right. \\
$
Therefore,
$
12 = 2 \times 2 \times 3 = {2^2} \times 3 \\
27 = 3 \times 3 \times 3 = {3^3} \\
$
LCM as we know is the product of the highest powers of the prime factors.
So applyingteh definition over her, the LCM of 12 and 27 will be:
$
= {2^2} \times {3^3} \\
= 4 \times 27 \\
= 108 \\
$
Now, with the LCM of 12 and 27 as 108, we will get back to solving the equation to find the value of x by subtraction. So the equation will now become:
$x = \dfrac{{12 \times 4 - 5 \times 9}}{{108}}$
Which will become:
$
x = \dfrac{{48 - 45}}{{108}} \\
\Rightarrow x = \dfrac{3}{{108}} \\
$
So, finally we will get the simplified value of ‘x’ as :
$ \Rightarrow x = \dfrac{1}{{36}}$
Hence,as per our assumption, x is the missing fraction, and $x = \dfrac{1}{{36}}$, therefore, the valueof the missing fraction will be $\dfrac{1}{{36}}$ .
Note: A cube and a cuboid are very similar having 6 faces. A cube is made up of 6 squares, hence all dimensions of a cube are equal, such that the volume of a cube of side a' units becomes ${a^3}$ while its total surface area is $6{a^2}$ square units.
So, to find the value of x, we will solve the given equation of fraction with the given mathematical operations in the equation.
Complete step-by-step answer:
Now, according to the equation given in the question, to find the missing fraction, let the missing fraction be x, then putting ‘x’ in the place of the missing fraction, the equation will change to:
$x + \dfrac{5}{{12}} = \dfrac{{12}}{{27}}$.
Now, on the LHS, x is being added to a fraction.
So, to separate x and solve for x, we have to subtract $\dfrac{5}{{12}}$ from both sides of the equation, so that we are left with just x on the LHS, such that it will be equal to the term on the RHS.
The other way to understand this is, we can simply take $\dfrac{5}{{12}}$to the other side, such that the + sign before $\dfrac{5}{{12}}$will change to a minus sign, when it is taken to the right hand side. So, the equation will now become:
$x = \dfrac{{12}}{{27}} - \dfrac{5}{{12}}$
So now we will subtract $\dfrac{5}{{12}}$ from $\dfrac{{12}}{{27}}$ . Now to do the subtraction on the RHS,
We have to first find out the LCM of 12 and 27, for which we need to find the prime factors of 12 and 27.
So prime factorization of 12 and 27 will give us the factors as:
$
2\left| \!{\underline {\,
{12} \,}} \right. \\
2\left| \!{\underline {\,
6 \,}} \right. \\
3\left| \!{\underline {\,
3 \,}} \right. \\
1\left| \!{\underline {\,
1 \,}} \right. \\
$ $
3\left| \!{\underline {\,
{27} \,}} \right. \\
3\left| \!{\underline {\,
9 \,}} \right. \\
3\left| \!{\underline {\,
3 \,}} \right. \\
1\left| \!{\underline {\,
1 \,}} \right. \\
$
Therefore,
$
12 = 2 \times 2 \times 3 = {2^2} \times 3 \\
27 = 3 \times 3 \times 3 = {3^3} \\
$
LCM as we know is the product of the highest powers of the prime factors.
So applyingteh definition over her, the LCM of 12 and 27 will be:
$
= {2^2} \times {3^3} \\
= 4 \times 27 \\
= 108 \\
$
Now, with the LCM of 12 and 27 as 108, we will get back to solving the equation to find the value of x by subtraction. So the equation will now become:
$x = \dfrac{{12 \times 4 - 5 \times 9}}{{108}}$
Which will become:
$
x = \dfrac{{48 - 45}}{{108}} \\
\Rightarrow x = \dfrac{3}{{108}} \\
$
So, finally we will get the simplified value of ‘x’ as :
$ \Rightarrow x = \dfrac{1}{{36}}$
Hence,as per our assumption, x is the missing fraction, and $x = \dfrac{1}{{36}}$, therefore, the valueof the missing fraction will be $\dfrac{1}{{36}}$ .
Note: A cube and a cuboid are very similar having 6 faces. A cube is made up of 6 squares, hence all dimensions of a cube are equal, such that the volume of a cube of side a' units becomes ${a^3}$ while its total surface area is $6{a^2}$ square units.