Question

Tyler used $1\dfrac{1}{4}$ gallons of paint to paint $\dfrac{5}{7}$ of a fence. How many gallons will it take to paint the whole fence?

Answer 1

Hint: We are given a word problem where we are given the amount of paint used in painting a fraction of fence and are asked to find the amount of total paint used in painting the whole fence. In order to solve this problem, we need to use a unitary method. The unitary method is a technique for solving a problem by first finding the value of a single unit and then finding the necessary value by multiplying the single unit from the value of a multiple. Since, here we just need the value of a single unit, we will only divide to obtain the solution.

Complete step by step answer:
(i)
The given information in the word problem is the amount of paint used to paint a part of a fence. Which is given as, $1\dfrac{1}{4}$ gallons of paint to paint $\dfrac{5}{7}$ of the fence.
We have to find the amount of paint required to paint the whole fence. Therefore, we will apply the unitary method to solve this problem.
Applying the unitary method, we know that:
Amount of paint required to paint $\dfrac{5}{7}$th part of the fence $ = $ $1\dfrac{1}{4}$ gallons of paint
(ii)
As we know that we can convert a mixed fraction to an improper fraction by multiplying the whole number with the denominator and adding it to the numerator and then writing the obtained value as the new numerator of the improper fraction and denominator being the same as previous denominator of the mixed fraction. Here, we will similarly change the mixed fraction $1\dfrac{1}{4}$ to an improper fraction.
So, we can write $1\dfrac{1}{4}$ as $\dfrac{5}{4}$ as we know that $\left( {4 \times 1} \right) + 1 = 5$.
Therefore, our statement becomes:
Amount of paint required to paint $\dfrac{5}{7}$th part of the fence $ = $ $\dfrac{5}{4}$ gallons of paint
(iii)
Continuing to apply the unitary method, we will find the amount of paint to paint the whole fence i.e., $1$ fence. Therefore,
Amount of paint required to paint the whole fence $ = $ $\dfrac{{\dfrac{5}{4}}}{{\dfrac{5}{7}}}$ gallons of paint
Since we know that division by a fraction is same as multiplication with its reciprocal. Therefore, we can say that,
Amount of paint required to paint the whole fence $ = $ $\dfrac{5}{4} \times \dfrac{7}{5}$ gallons of paint
(iv)
Solving the expression $\dfrac{5}{4} \times \dfrac{7}{5}$ by cancelling the common factor $5$ from the numerator and the denominator, we will get:
Amount of paint required to paint the whole fence $ = $ $\dfrac{7}{4}$ gallons of paint

Hence, the amount of paint required to paint the whole fence is $\dfrac{7}{4}$ gallons of paint or $1.75$ gallons of paint.

Note: Always remember to first convert the mixed fraction to improper fraction before starting to solve the problem as a mixed fraction cannot be directly multiply or divide with proper or improper fraction. It first needs to be converted into an improper fraction. If we were asked to calculate the amount of paint required to paint multiple fences i.e., more than one fence, we would have directly multiplied the number of fences with the amount of paint required to paint one fence to obtain the answer.