Question

How many \[kg\] of mixed nuts that contain \[40\% \] peanuts must Jenny add to \[6kg\] of mixed nuts that contain \[30\% \] peanuts to make a mixture with\[34\% \] peanuts?

Answer 1

Hint: Here, we will use the percentage formula to find the mass of Peanuts in both the bags. Then, by adding the obtained masses, we will find the total mass of peanuts. We will find the total mass of the bag by adding the masses of both the bags. By using the percentage of peanuts in the mixture, we will get a linear equation and by solving the linear equation, we will find the mass added to the mixture and hence the required answer.

Formula Used:
Amount of quantity is given by \[x = \dfrac{P}{{100}} \times n\] where \[x,P,n\] are the amount of quantity, percentage and the total amount respectively.

Complete Step by Step Solution:
We are given \[6kg\] of mixed nuts which contain \[30\% \] peanuts.
Amount of quantity is given by \[x = \dfrac{P}{{100}} \times n\] .
Now, we will find the quantity of peanuts in the first bag of mixed nuts, by using the Percentage formula, we get
Mass of Peanuts \[ = \dfrac{{30}}{{100}} \times 6{\rm{ }}kg\]
Now, by simplifying the terms, we get
\[ \Rightarrow \] Mass of Peanuts \[ = 0.3 \times 6{\rm{ }}kg\]
Multiplying the terms, we get
\[ \Rightarrow \] Mass of Peanuts \[ = 1.8{\rm{ }}kg\]
Let \[x\] be the kilograms of mixed nuts that contain \[40\% \] peanuts.
Now, we will find the quantity of peanuts in the second bag of mixed nuts, by using the Percentage formula, we get
Mass of Peanuts \[ = \dfrac{{40}}{{100}} \times x{\rm{ }}kg\]
Now, by simplifying the terms, we get
\[ \Rightarrow \] Mass of Peanuts \[ = 0.4x{\rm{ }}kg\]
Now, the total mixture of Peanuts in the bag is by adding the mass of peanuts. Therefore, we get
\[ \Rightarrow \] Total mass of Peanuts \[ = \left( {0.4x + 1.8} \right)kg\]……………………………….\[\left( 1 \right)\]
Now, we will find the mass of the total bag by adding the mass added to the second bag and the mass of the first bag.
\[ \Rightarrow {m_{total}} = \left( {x + 6} \right)kg\] …………………………………………\[\left( 2 \right)\]
We are given that the new mixture contains \[34\% \] peanuts.
Now, we will find the mass of the mixture containing \[34\% \] peanuts, we get
\[ \Rightarrow \dfrac{{{m_{p}}}}{{{m_{t}}}} = \dfrac{{34}}{{100}}\]
\[ \Rightarrow \dfrac{{{m_{p}}}}{{{m_{t}}}} = 0.34\] …………………………………………………………...\[\left( 3 \right)\]
Now, by substituting the equations \[\left( 1 \right)\] and\[\left( 2 \right)\] in equation \[\left( 3 \right)\], we get
\[ \Rightarrow \dfrac{{0.4x + 1.8}}{{x + 6}} = 0.34\]
Now, by rewriting the equation, we get
\[ \Rightarrow 0.4x + 1.8 = \left( {x + 6} \right)0.34\]
Now, by multiplying the terms, we get
\[ \Rightarrow 0.4x + 1.8 = 0.34x + 2.04\]
Now, by rewriting the equation, we get
\[ \Rightarrow 0.4x - 0.34x = 2.04 - 1.8\]
Now, by simplifying the equation, we get
\[ \Rightarrow 0.06x = 0.24\]
Now, by multiplying by 100, we get
\[ \Rightarrow 6x = 24\]
Now, by rewriting the equation, we get
\[ \Rightarrow x = \dfrac{{24}}{6}\]
\[ \Rightarrow x = 4kg\]

Therefore, Jenny added to 4 kg of mixed nuts that contain \[40\% \] peanuts.

Note:
We know that since the quantity is in percentage, then the total quantity would be always 100. We know that a linear equation is defined as an equation with the highest degree as one. Linear equations are a combination of constants and variables. Constants are the numbers whereas variables are represented in letters. We should also know that every linear equation in one variable has a one and unique solution. We can solve the linear equation easily.