Josh and Marcia bought a number of chairs at an auction. The average

Let the number of chairs purchased by Josh is a.
Let the number of chairs purchased by Marcia is b.

50b+62a=53 (a+b)
50b + 62a = 53a + 53b
9a = 3b
so, \(\frac{b}{a}\) = \(\frac{9}{3}\) = \(\frac{3}{1}\)

IMO the answer is D.

Use allegation method for weighted averages

Josh's Average----------------------------Overall Average--------Marcia's Average
------|----------------------------------------------------------------------|----------
------|----------------------------------------------------------------------|----------
---No. of Josh's chairs----------------------------------------------No. of Marcia's chairs

-----62-----------------------------------------------53------------------50---------
------|----------------------------------------------------------------------|----------
------|----------------------------------------------------------------------|----------
---No. of Josh's chairs----------------------------------------------No. of Marcia's chairs

\(\frac{No. of Marcia's Chairs}{No. of Josh's Chairs} = \frac{62-53}{53-50} = \frac{9}{3} = \frac{3}{1}\)

Answer is (D)

This is a straightforward question on alligation. Drawing the alligation diagram will help you solve this question in a jiffy.

The alligation diagram can be drawn as shown in the figure below:


From the diagram, it’s clear that the ratio of the number of chairs purchased by Marcia and Josh is 3:1. The correct answer option is D.

Alternatively, you can use the weighted average concept to arrive at the same answer, since the alligation diagram is nothing but an alteration of the weighted average equation.
Hope this helps!

let price chair of Josh Pj and Marcia Pm
total chair of Josh Cj and Marchia Cm
need to find Cm/Cj
given
50=Pj/Cj
Pm/cm=62
also
Pj+Pm/Cj+cm=53
so
we can say
50cj+62cm=53cj+53cm
we get Cj/Cm = 3:1
IMO D

Solution

Given:

• Josh and Marcia bought a number of chairs at an auction.
• Average price of the chairs Marcia purchased = $50
• Average price of the chairs Josh purchased = $62
• Average price of all the chairs purchased = $53

To find:

• The ratio of the number of chairs purchased by Marcia to the no. purchased by Josh

Approach and Working

Method -1) weighted Average


We are given that average price of the chairs Marcia purchased = $50

• Therefore, Price of all the chairs bought by Marcia/ Number of chairs purchased by Marcia = $ 50

o Let us assume that total number of chairs purchased by Marcia = M
• Hence, Price of all the chairs bought by Marcia/ M = $ 50

Price of all the chairs bought by Marcia = $ 50 M

We are also given that average price of the chairs Josh purchased = $62

• Therefore, Price of all the chairs bought by Josh / Number of chairs purchased by Josh = $ 62

o Let us assume that total number of chairs purchased by Josh = J
• Hence, Price of all the chairs bought by Josh / J = $ 62

Price of all the chairs bought by Josh = $ 62 J

Next, we have the average price of all the chairs purchased = $53

• Therefore, Price of all the chairs bought / Number of chairs purchased = $ 53
• ($ 50 M + $62 J)/ (M + J) =$ 53

o $ 50 M + $62 J = $ 53 (M + J)
o $ 50 M + $62 J = $ 53 M + $ 53 J
o 9 J = 3M
o 3 J = M
o M: J = 3: 1

Hence, the correct answer is D.

Correct answer: Option D

Method -2) Alligation



M: J = (62-53) : (53 – 50) = 9 : 3 = 3:1

Hence, the correct answer is D.

Correct answer: Option D