Question

If 2kg of metal of which \[\dfrac{1}{3}\] is zinc and rest is copper be mixed with 3kg of metal of which \[\dfrac{1}{4}\] is zinc and rest is copper, what is the ratio of zinc to copper in the mixture?
A.\[13:35\]
B. \[12:30\]
C. \[15:40\]
D. \[17:43\]

Answer 1

Hint: We calculate the part which is copper in the 2kg metal by subtracting given part value of zinc from 1. Similarly, calculate the part which is copper in the 3kg metal by subtracting given part value of zinc from 1. Write the total quantity of zinc from both metals and total quantity of copper from both metals. Calculate the ratio of zinc to copper.
* Ratio \[m:n\]can be written in the form of fraction as \[\dfrac{m}{n}\]

Complete step-by-step answer:
We are given two metals. Calculate the quantity of zinc and copper in each metal separately.
Metal 1:
We have total weight of metal as 2kg
Since \[\dfrac{1}{3}\]of metal is zinc then we can write \[\left( {1 - \dfrac{1}{3}} \right)\] of metal will be copper
\[ \Rightarrow \dfrac{1}{3}\]of metal is zinc and \[\dfrac{2}{3}\]of metal is copper. … (1)
Metal 2:
We have total weight of metal as 3kg
Since \[\dfrac{1}{4}\]of metal is zinc then we can write \[\left( {1 - \dfrac{1}{4}} \right)\] of metal will be copper
\[ \Rightarrow \dfrac{1}{4}\]of metal is zinc and \[\dfrac{3}{4}\]of metal is copper. … (2)
Now we can write the value of total quantity of zinc and copper from equations (1) and (2)
Total Zinc:
\[ \Rightarrow \]Total Zinc in the mixture \[ = \dfrac{1}{3} \times (2) + \dfrac{1}{4} \times (3)\]
\[ \Rightarrow \]Total Zinc in the mixture \[ = \dfrac{2}{3} + \dfrac{3}{4}\]
Take LCM of the fractions
\[ \Rightarrow \]Total Zinc in the mixture \[ = \dfrac{{2 \times 4 + 3 \times 3}}{{4 \times 3}}\]
\[ \Rightarrow \]Total Zinc in the mixture \[ = \dfrac{{8 + 9}}{{12}}\]
\[ \Rightarrow \]Total Zinc in the mixture \[ = \dfrac{{17}}{{12}}\] … (3)
Total Copper:
\[ \Rightarrow \]Total Copper in the mixture \[ = \dfrac{2}{3} \times (2) + \dfrac{3}{4} \times (3)\]
\[ \Rightarrow \]Total Copper in the mixture \[ = \dfrac{4}{3} + \dfrac{9}{4}\]
Take LCM of the fractions
\[ \Rightarrow \]Total Copper in the mixture \[ = \dfrac{{4 \times 4 + 9 \times 3}}{{4 \times 3}}\]
\[ \Rightarrow \]Total Copper in the mixture \[ = \dfrac{{16 + 27}}{{12}}\]
\[ \Rightarrow \]Total Copper in the mixture \[ = \dfrac{{43}}{{12}}\] … (4)
Now we have to calculate the ratio of total zinc to the total copper in the mixture.
Since we know ratio \[m:n\]can be written in the form of fraction as \[\dfrac{m}{n}\]
We write Zinc : Copper \[ = \dfrac{{\dfrac{{17}}{{12}}}}{{\dfrac{{43}}{{12}}}}\]
Write the fraction in simpler form
\[ \Rightarrow \]Zinc : Copper \[ = \dfrac{{17}}{{12}} \times \dfrac{{12}}{{43}}\]
Cancel same factors from numerator and denominator
\[ \Rightarrow \]Zinc : Copper \[ = \dfrac{{17}}{{43}}\]
\[ \Rightarrow \]Zinc : Copper \[ = 17:43\]

\[\therefore \]Option D is correct.

Note:
Many students make the mistake of assuming the part alone as the value of quantity and don’t multiply the part to the quantity to obtain the quantity which is wrong, keep in mind we are given that metal has this part as zinc and remaining part as copper. Also, keep in mind ratio should always be in simplest form i.e. there should not be any common factor between numerator and denominator.