Question

Paulette, Quentin, and Rosie each donated money to a charity. Paulette gave as much money as Quentin and Rosie gave together. If Quentin had given three times more than he gave, he would have given $ 40 $ more than Paulette. And if Rosie had given $ 20 $ less, she would have given half as much as Paulette. How much did Paulette give?
A. $ 80 $
B. $ 120 $
C. $ 160 $
D. $ 200 $

Answer 1

Hint: Here we will first convert the given word statements in the form of the mathematical expressions. Then we will substitute the values accordingly, simplify and will find the required unknowns.

Complete step-by-step answer:
Let us consider that the amount Paulette, Quentin and Rosie donated to a charity be a, b and c.
Paulette gave as much money as Quentin and Rosie gave together.
 $ \Rightarrow a = b + c\;{\text{ }}....{\text{ (i)}} $
If Quentin had given three times more than he gave, he would have given $ 40 $ more than Paulette.
 $
   \Rightarrow 3b = a + 40 \\
   \Rightarrow b = \dfrac{{a + 40}}{3}\;{\text{ }}.....{\text{ (ii)}} \\
  $
Also, given that if Rosie had given $ 20 $ less, she would have given half as much as Paulette.
 $
  c - 20 = \dfrac{a}{2} \\
   \Rightarrow c = \dfrac{a}{2} + 20\;{\text{ }}.....{\text{ (iii)}} \;
  $
Substitute the values of the equation (iii) and (ii) in the equation (i)
 $ a = \dfrac{{a + 40}}{3} + \dfrac{a}{2} + 20 $
Take LCM and simplify the above equation –
 $ a = \dfrac{{2\left( {a + 40} \right)}}{{3(2)}} + \dfrac{{3a}}{{2(3)}} + 6\dfrac{{(20)}}{6} $
 $ \Rightarrow a = \dfrac{{2a + 80 + 3a + 120}}{6} $
Simplify among the like terms-
 $
   \Rightarrow a = \dfrac{{\underline {2a + 3a} + \underline {80 + 120} }}{6} \\
   \Rightarrow a = \dfrac{{5a + 200}}{6} \;
  $
Do-cross multiplication –
 $ \Rightarrow 6a = 5a + 200 $
Move the like terms on one side of the equation. When the term is moved from one side to another, the sign of the term also changes. Positive term changes to negative and vice-versa.
 $
   \Rightarrow 6a - 5a = 200 \\
   \Rightarrow a = 200 \;
  $
Thus, Paulette donated $ 200\ $
So, the correct answer is “200”.

Note: Always convert the given word statements in the proper form of the expressions. Check it twice as the solution further solely depends on it. Be good in LCM and simplify the equations for the efficient and accurate solution.