Grades 9-10 Video Solutions 2014-Levels 9&10 Video Solutions 2014 problem19

Levels 9&10 Video Solutions 2014 problem19

Video Transcription

Question number 19. Triplets Jane, Danielle, and Hannah wanted to buy identical hats. However, Jane lacked a third of the price, Danielle a quarter, and Hannah a fifth.  When the hats became $9.40 cheaper, the sisters combined their money and each of them bought a hat. Not a cent was left. What was the price of one hat before the price reduction?  So let that number, the price of the hat before the price reduction, be X.  Let me just say that's the price before the sale. And what we have is three hats are being purchased. And for a price of X minus $9.40 each. So that's how much money is being paid.  And that is equal to the combined money between the three sisters. However, Jane lacked a third of the price, so she only had two thirds of the price.  Danielle a quarter of the price, so she had three fourths of the price. And Hannah a fifth, so she had four fifths available. So this is the equation we have to solve for X.  OK, so on the right hand side, what we can do is obtain a common denominator. And that common denominator is 60.  So two thirds, we multiply by 20 on top and bottom and obtain 40 here. For four, we multiply by 15, obtain a 45. And for five, we multiply by 12. So we obtain 48.  And so what we have here is 133 divided by 60 times X. Now we can divide both sides by three. And we have X minus $9.40 is equal to 133 divided by 180. That's 60 times 3.  And that's equal to X. So continuing on, we can collect X's on one side of the equation. Let me do that here in the next column. So we have X minus 133 over 180.  X is equal to $9.40. So on the left hand side, we have 47 over 180. X on the right hand side, let me write this as a fraction, 94 divided by 10. And then we can keep solving.  So X has to be 94 divided by 10 times 180 divided by 47. And then when we do that, simplifying a little bit, 94 times 18, the 10 cancels with the 180 divided by 47.  We can either come up with prime factors or compute in some other way. The answer here comes out to 36. And that is the price of the hat in dollars before the price was reduced.  That comes out to the $36.

Video Summary

Triplets Jane, Danielle, and Hannah each wanted to buy the same hat but faced different financial constraints: Jane was missing a third, Danielle a quarter, and Hannah a fifth of the hat's original price. After a $9.40 reduction in price, they pooled their money and bought the hats with no money left. Solving for the original price involved calculating the combined fractional money they had and forming an equation. By simplifying the equation, the original price of one hat is determined to be $36.