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Video: Calculating Areas by Multiplying Mixed Numbers in Word Problems

Tim Burnham

The dimensions of a wall are 8 5/8 feet and 20 1/5 feet. If a gallon of paint can cover about 230 square feet, will one gallon of paint be enough to cover the wall?

06:38

Video Transcript

The dimensions of a wall are eight and five-eighths feet and twenty and one-fifth feet. If a gallon of paint can cover two hundred and thirty square feet, will one gallon of paint be enough to cover the wall?

Well I always like to draw a little diagram to help me think about these problems, and here we have a little wall and it’s twenty and one-fifth feet wide and eight and five-eighths feet high.

Now if we’ve got one gallon of paint in order to paint the wall and a pat- and one gallon of paint can cover two hundred and thirty square feet, then what this question is really saying is, is the area of the wall less than or equal to two hundred and thirty square feet? Now to work out the area of the wall, remember it’s a rectangle, so we can do the length times the height.

So that’s twenty and one-fifth times eight and five-eighths, so now we’re multiplying two mixed numbers together. And the best way of completing this calculation is to convert those mixed numbers into top heavy fractions or to improper fractions.

Now remember twenty and a fifth really means twenty plus one-fifth and eight and five-eighths means eight plus five-eighths. And we can rewrite twenty and eight as twenty over one and eight over one to turn them into fractions.

So twenty over one is also the same as twenty and eight over one is also the same as eight. Now we’ve got some fraction addition to do within those parentheses, so twenty over one plus one over five, but the problem with that is that they don’t have common denominators. So I need to multiply twenty over one by some version of one so that it would have a denominator of five like a fifth, and the best way of doing that is to multiply by five over five.

And likewise for eight over one, I need a common denominator with five-eighths, so I need a denominator of eight. So if I multiply by eight over eight, which after all is just the same as one, then I’ll have a denominator of eight.

So twenty and a fifth is the same as a hundred over five plus one over five and eight and five-eighths is the same as sixty-four over eight plus five over eight. And a hundred over five plus one over five is a hundred and one over five, and sixty-four over eight plus five over eight is sixty-nine over eight.

Now looking at those, unfortunately nothing cancels, so we’re just gonna have to do the straight multiplication. A hundred and one times sixty-nine is the same as a hundred times sixty-nine plus sixty-nine, and five times eight is forty. Well a hundred times sixty-nine is six nine zero zero, and six nine zero zero, six thousand nine hundred, plus sixty-nine is six nine six nine. So that’s six nine six nine, well six thousand nine hundred and sixty-nine, over forty.

So that’s six thousand nine hundred and sixty-nine divided by forty, so we now need to do this calculation. What is forty into six nine six nine? Or forties into sixty-nine just go once.

And one lot of forty is forty. Now the difference between sixty-nine and forty-nine take away nothing is nine, and six take away four is two. Now we can bring down the six, and we’re looking for forties into two nine six.

Well seven times four is twenty-eight, so seven times forty is gonna be two hundred and eighty; eight times forty will be too big, so seven. And as we said seven times forty is two hundred and eighty.

Now working out the difference between two hundred and ninety-six and two hundred and eighty-six take away nothing is six, nine take away eight is one, and two take away two is nothing. So let’s bring down the nine.

And we need to know how many forties go into one hundred and sixty-nine. Well four times four is sixteen, so four times forty would be a hundred and sixty. Five times forty would be two hundred; that’d be too big, so that’s gonna be four.

And four times forty as we said is a hundred and sixty. So the remainder is going to be nine and we can bring down the next zero and so on. So forties into ninety go twice, so we don’t need to know any more than that. We’ve got a rough answer, a hundred and seventy-four point two something; let’s just round it to a hundred and seventy-four.

We wanted to know was the area of the wall less than or equal to two hundred and thirty square feet. And it clearly is. A hundred and seventy-four is definitely less than two hundred and thirty, so our one gallon of paint is definitely gonna be enough to do the wall.

Now remember just giving our answer of a hundred and seventy-four square feet wouldn’t be enough to give you full marks on this question. The question says, will one gallon of paint be enough to cover the wall? So you have to have- actually have to answer that and say yes.

But also remember you do need to have the calculations to back up your answer, so you need the combination of the calculations and the specific answer to the question. Now we took a very precise approach to that question; we could have done a slightly more approximate method.

Now we could’ve said, instead of eight and five-eighths feet, let’s say that’s approximately nine feet. So we rounded up to the next number of feet.

And instead of twenty and a fifth feet, let’s round that up to twenty-one feet. So we’ve overestimated the length and the height of our wall. So if we calculate the area of this overestimated wall size, we just have to do the calculation twenty-one times nine, which of course is twenty plus one times nine. And then we can use the distributive property of multiplication to say that that’s twenty times nine plus one times nine. So that’s a hundred and eighty plus nine.

So our overestimate for the area is a hundred and eighty-nine square feet; we know that the actual area of the wall is less than a hundred and eighty-nine square feet, and that is also clearly less than two hundred and thirty square feet.

So without having to go into excruciating detail on very difficult calculations like we did last time, we’ve got some simple rounded numbers. We know it’s an overestimate; we still easily got enough paint, so we know that yes we do have enough paint in one gallon to cover the wall.