Question

Add $\dfrac{1}{4}$ to the sum of $\dfrac{3}{5}$ and $\dfrac{7}{9}$ and subtract it from $\dfrac{7}{27}$. What is the final value?

Answer 1

Hint: We will convert the given word problem into a mathematical expression. Then we will simplify the obtained expression. To simplify the obtained expression, we will use the precedence of the brackets, that is, we will solve the expression inside the innermost brackets first. After solving the equation in this way, we will get the final value.

Complete step-by-step solution
The given statement in the problem is the following,
add $\dfrac{1}{4}$ to the sum of $\dfrac{3}{5}$ and $\dfrac{7}{9}$ and subtract it from $\dfrac{7}{27}$.
So, we will first take the sum of $\dfrac{3}{5}$ and $\dfrac{7}{9}$. Our expression is $\dfrac{3}{5}+\dfrac{7}{9}$. Next, we have to add $\dfrac{1}{4}$ to this sum. So now our expression becomes $\left( \dfrac{3}{5}+\dfrac{7}{9} \right)+\dfrac{1}{4}$. Next, we have to subtract this value from $\dfrac{7}{27}$. So, we get the following expression,
$\dfrac{7}{27}-\left( \left( \dfrac{3}{5}+\dfrac{7}{9} \right)+\dfrac{1}{4} \right)$
Now, we will start solving the above expression. First we will solve the expression inside the innermost bracket. For that, we will first take the LCM of the denominator as follows,
$\dfrac{3}{5}+\dfrac{7}{9}=\dfrac{9\times 3+7\times 5}{5\times 9}$
Simplifying the above expression, we get
$\dfrac{3}{5}+\dfrac{7}{9}=\dfrac{9\times 3+7\times 5}{5\times 9}=\dfrac{27+35}{45}=\dfrac{62}{45}$
We will substitute this value in place of the innermost bracket as follows,
$\dfrac{7}{27}-\left( \dfrac{62}{45}+\dfrac{1}{4} \right)$
Now, we will solve the next bracket. For this as well, we will take the LCM of the denominator as follows,
$\dfrac{62}{45}+\dfrac{1}{4}=\dfrac{62\times 4+45\times 1}{45\times 4}$
Simplifying this expression, we get
$\dfrac{62}{45}+\dfrac{1}{4}=\dfrac{62\times 4+45\times 1}{45\times 4}=\dfrac{248+45}{180}=\dfrac{293}{180}$
Substituting this value in place of the expression in the bracket, we get the following expression,
$\dfrac{7}{27}-\dfrac{293}{180}$
Again, we will take the LCM of the denominators and then simplify in the following manner,
 $\dfrac{7}{27}-\dfrac{293}{180}=\dfrac{7\times 20-293\times 3}{540}=\dfrac{140-879}{540}=\dfrac{-739}{540}$
Therefore, the final value obtained is $-\dfrac{739}{180}$.

Note: In such types of questions, it is important that we obtain a correct mathematical expression from the statement given in the question. The key aspect of obtaining the final value is the precedence given to the expressions inside the innermost brackets. There is a possibility of making minor errors in the calculations of such questions. to avoid these errors, it is useful to write the calculations explicitly.