Understanding the Groundnut Oil Mixture Problem

The problem asks us to find out how much pure groundnut oil needs to be added to a given amount of impure groundnut oil mixture to increase its purity percentage.

We start with a certain quantity of groundnut oil mixture that is 80% pure. We want to add 100% pure groundnut oil to this mixture so that the final mixture becomes 95% pure.

Initial Mixture Analysis

We are given 20 kgs of 80% pure groundnut oil mixture.

• Total weight of initial mixture = 20 kg
• Percentage of pure groundnut oil in initial mixture = 80%
• Percentage of impurity in initial mixture = 100% - 80% = 20%

Let's calculate the actual weight of pure oil and impurity in the initial mixture:

• Weight of pure groundnut oil initially = $80\%$ of $20$ kg $= \frac{80}{100} \times 20 = 0.80 \times 20 = 16$ kg
• Weight of impurity initially = $20\%$ of $20$ kg $= \frac{20}{100} \times 20 = 0.20 \times 20 = 4$ kg

We can verify this: $16$ kg (pure oil) + $4$ kg (impurity) = $20$ kg (total mixture).

Adding Pure Groundnut Oil

We are adding pure groundnut oil to this mixture. Let's say we add $x$ kilograms of pure groundnut oil.

Pure groundnut oil is 100% pure. This means adding $x$ kg of pure oil increases only the pure oil content and the total weight of the mixture.

Final Mixture Analysis

After adding $x$ kgs of pure groundnut oil:

• Weight of pure groundnut oil in the final mixture = (Initial pure oil) + (Added pure oil) $= 16 + x$ kg
• Weight of impurity in the final mixture = (Initial impurity) $= 4$ kg (because only pure oil was added, the impurity amount remains unchanged)
• Total weight of the final mixture = (Initial total weight) + (Added pure oil) $= 20 + x$ kg

The problem states that the resulting mixture must contain 95% pure groundnut oil.

This means:

$\text{Percentage of pure oil in final mixture} = \frac{\text{Weight of pure oil in final mixture}}{\text{Total weight of final mixture}} \times 100\%$

$95\% = \frac{16 + x}{20 + x} \times 100\%$

Solving for the Amount of Pure Groundnut Oil Added

Now, we set up the equation and solve for $x$:

$\frac{16 + x}{20 + x} \times 100 = 95$

Divide both sides by 100:

$\frac{16 + x}{20 + x} = \frac{95}{100}$

Simplify the fraction $\frac{95}{100}$ by dividing both numerator and denominator by 5:

$\frac{95 \div 5}{100 \div 5} = \frac{19}{20}$

So the equation becomes:

$\frac{16 + x}{20 + x} = \frac{19}{20}$

Now, cross-multiply:

$20 \times (16 + x) = 19 \times (20 + x)$

Distribute the numbers on both sides:

$(20 \times 16) + (20 \times x) = (19 \times 20) + (19 \times x)$

$320 + 20x = 380 + 19x$

Now, gather the $x$ terms on one side and the constant terms on the other side. Subtract $19x$ from both sides:

$320 + 20x - 19x = 380 + 19x - 19x$

$320 + x = 380$

Subtract 320 from both sides:

$320 + x - 320 = 380 - 320$

$x = 60$

So, 60 kilograms of pure groundnut oil should be added.

Verification

Let's check the final mixture with $x = 60$ kg:

• Pure oil in final mixture = $16 + 60 = 76$ kg
• Impurity in final mixture = $4$ kg
• Total weight of final mixture = $20 + 60 = 80$ kg

Percentage purity in the final mixture = $\frac{\text{Pure oil}}{\text{Total mixture}} \times 100\% = \frac{76}{80} \times 100\%$

$\frac{76}{80} = \frac{76 \div 4}{80 \div 4} = \frac{19}{20}$

Percentage purity = $\frac{19}{20} \times 100\% = 19 \times \frac{100}{20}\% = 19 \times 5\% = 95\%$

This matches the requirement of the problem, confirming our answer is correct.

The calculation shows that 60 kg of pure groundnut oil must be added.