Video: Solving Real-World Problems Including Rational Expressions

It takes Victoria 2 hours to pick up 8 kg of strawberries. When she is helped by her son, Ethan, it takes them 1 hour and a quarter. How long would it take Ethan to pick up 8 kg of strawberries alone? Give your answer in hours and minutes.

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Video Transcript

It takes Victoria two hours to pick up eight kilograms of strawberries. When she is helped by her son Ethan it takes them one hour and a quarter. How long would it take Ethan to pick up eight kilograms of strawberries alone? Give your answer in hours and minutes.

When two people are working together on a task, we can create this equation: one divided by 𝐴 plus one divided by 𝐵 equals one divide by the total where 𝐴 is the amount of time it takes one person to do a job, 𝐵 would be the amount of time it takes another person to do a job, and then the total would be how long it takes them to do it together.

So the plus sign presents them working together. So we can let 𝐴 represent Victoria and 𝐵 represent Ethan. Victoria and her son Ethan are working together to pick up eight kilograms of strawberries. And we know how long it takes Victoria. It takes Victoria two hours to pick up eight kilograms of strawberries. We do not know how long it takes Ethan to pick up the strawberries.

We know when she is helped by her son Ethan that it takes them one hour and a quarter. So that’s the total it would take them together. So the total can be replaced with 1.25, one plus a quarter, 1.25. So we want to solve for 𝐵, the amount of time that would take Ethan to pick up eight kilograms of strawberries by himself. So in order to solve for 𝐵, we need to take the one-half fraction and bring it over to the right-hand side of the equation.

So we need to subtract one-half from both sides of the equation. The one-halves cancel on the left. And then on the right, we need to subtract these fractions. But in order to do so, we need to have common denominators. But first, we don’t wanna have a decimal inside of the fraction. So how can we rewrite 1.25? Well, 0.25 represents a quarter. And a quarter, instead of writing it as 0.25, we could represent it by one-fourth.

So one and one-fourth could be four-fourths plus one-fourth. And four-fourths plus one-fourth would be five-fourths because we add the numerators and keep the common denominator. Now one divided by five-fourths means we need to flip our fraction on the bottom, because when dividing fractions, we multiply by their reciprocals. So we need to take one times four-fifths and one times four-fifths is four-fifths.

So we have one over 𝐵 is equal to four-fifths minus one- half. So as we said before, in order to add or subtract fractions, we have to have a common denominator. So what is a number that five and two can both go into? It would be 10. So how would we go from five to 10? What will we multiply by? We have to multiply by two. So if we multiply the denominator by two, we’ll need to multiply the numerator by two. And four times two is eight.

Now for the one-half, to get from two to 10, we multiply by five. So we need to do the same to the numerator, and one times five is five. So we have eight-tenths minus five- tenths. So we keep our common denominator of 10 and we’d subtract the numerators. So we have that one over 𝐵 is equal to three-tenths. So in order to solve for 𝐵, we can cross multiply.

We take 𝐵 times three and we set it equal to one times 10, which is 10. So to solve for 𝐵, we need to divide both sides of the equation by three. So 𝐵 is equal to ten-thirds. So how can we rewrite ten-thirds if we’re asked to write in hours and minutes? Well, ten-thirds is the same as three and one-third. So three and one-third means that it’s three hours and then one-third minutes.

Well what does the one-third represent? It means it’s one-third of an hour because three is an hour so it’s three hours and one-third an hour. So what is one-third of an hour? Well If we take 60 minutes and split into three, it would be 20 minutes plus 20 minutes plus 20 minutes or simply rewriting it as three and twenty sixtieths, so 20 minutes out of 60 minutes.

Therefore, the time that it would take Ethan to do this alone would be three hours and 20 minutes.

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