Noah constructs three-fourths of a wall in one and two-thirds days. How many days will he need to construct the wall?
We can let this rectangle represent the whole wall. And then we can divide it in fourths. We know that Noah has constructed three-fourths of the wall. And to construct this three-fourths of the wall, it took one and two-thirds days. We want to know how long it would take to construct the whole wall.
To do that, we’ll need to know how much time it would take to construct the final one-fourth. And that means we’ll need to do some division. If it took one and two-thirds days to construct three-fourths of the wall, to find out how much time each fourth took, you would need to divide the three-fourths time by three, one and two-thirds divided by three. But we don’t wanna work with this mixed number one and two-thirds. So we can rewrite it as five-thirds. This is because if we rewrite the whole number one with the denominator of three, it will be three over three. And then we add two-thirds to three-thirds, and we get five-thirds.
Okay, so now we know that we need to divide five-thirds by three. But since we’re working with fractions, to do that division, we’ll instead multiply by the reciprocal. The reciprocal of three is one-third. And so we can say that five-thirds divided by three is equal to five-thirds times one-third. Then we multiply the numerators — five times one is five — multiply the denominators — three times three is nine — to get five-ninths. This tells us that each fourth of the wall took five-ninths of a day.
And that means to find the total time, we can take one and two-thirds and add five-ninths to it. Again, we’ll want to work with an improper fraction instead of a mixed number. We want to add five-thirds plus five-ninths. Be careful here, just because we have the same numerators does not mean we can add these fractions.
Remember, to add fractions, we need a common denominator. If we multiplied five-thirds by three over three, in the numerator, five times three is 15. And in the denominator, three times three is nine, so that we’ll add fifteen-ninths plus five-ninths. And that would give us twenty-ninths.
In this case, we might want to rewrite twenty-ninths as a mixed number. To do that, we divide 20 by nine. Nine goes into 20 two times. Two times nine is 18, with a remainder of two. To write this out, we take the whole number, the number of times the nine goes into 20, and that is the whole number piece of our mixed number. And then we take the remainder, which becomes the numerator of the fraction piece of our mixed number. And the divisor is the denominator of the fractional part of the mixed number. And that means we can say that it took two and two-ninth days to finish the whole wall.