49 Kg of blended tea contains Assam and Darjeeling tea in the ratio 5 : 2. Then the quantity of Darjeeling tea is to be added to the mixture to make the ratio of Assam to Darjeeling tea 2 : 1 is:
3.5 kg
Solving Tea Mixture Ratio Problems
This problem involves adjusting the ratio of two types of tea in a mixture by adding more of one type. We start with a total quantity of blended tea and its initial ratio, and we want to find out how much of one component needs to be added to achieve a new desired ratio.
Initial Quantity of Teas
The total quantity of blended tea is 49 kg. The ratio of Assam tea to Darjeeling tea is 5 : 2.
To find the initial quantity of each type of tea, we first find the total number of ratio parts:
Total ratio parts = \(5 + 2 = 7\)
Now we can calculate the weight of Assam tea and Darjeeling tea:
| Tea Type | Ratio Part | Calculation | Quantity (kg) |
|---|---|---|---|
| Assam Tea | 5 | \(\frac{5}{7} \times 49\) | 35 |
| Darjeeling Tea | 2 | \(\frac{2}{7} \times 49\) | 14 |
So, initially, the mixture contains 35 kg of Assam tea and 14 kg of Darjeeling tea.
Adding Darjeeling Tea to Change Ratio
We are adding a certain quantity of Darjeeling tea to the mixture to make the new ratio of Assam tea to Darjeeling tea 2 : 1.
Let the quantity of Darjeeling tea added be \(x\) kg.
- The quantity of Assam tea remains unchanged: 35 kg
- The new quantity of Darjeeling tea will be: \(14 + x\) kg
The new ratio of Assam tea to Darjeeling tea is given as 2 : 1. We can set up an equation based on this ratio:
\[ \frac{\text{Quantity of Assam Tea}}{\text{New Quantity of Darjeeling Tea}} = \frac{2}{1} \] \[ \frac{35}{14 + x} = \frac{2}{1} \]Solving for the Quantity Added
Now we solve the equation for \(x\):
Cross-multiply the equation:
\[ 35 \times 1 = 2 \times (14 + x) \] \[ 35 = 28 + 2x \]Subtract 28 from both sides:
\[ 35 - 28 = 2x \] \[ 7 = 2x \]Divide by 2 to find \(x\):
\[ x = \frac{7}{2} \] \[ x = 3.5 \]So, 3.5 kg of Darjeeling tea is to be added to the mixture.
Checking the New Mixture
If 3.5 kg of Darjeeling tea is added:
- Assam tea: 35 kg
- New Darjeeling tea: \(14 + 3.5 = 17.5\) kg
New ratio of Assam : Darjeeling = \(35 : 17.5\)
To simplify the ratio, divide both parts by 17.5:
\[ \frac{35}{17.5} = 2 \] \[ \frac{17.5}{17.5} = 1 \]The new ratio is 2 : 1, which matches the required ratio.
Therefore, the quantity of Darjeeling tea to be added is 3.5 kg.
Revision Table: Steps to Solve Ratio Problems
| Step | Action | Application (This Problem) |
|---|---|---|
| 1 | Calculate initial quantities using the given ratio and total amount. | Found 35 kg Assam, 14 kg Darjeeling in 49 kg total. |
| 2 | Identify which component is being changed and by how much (let it be 'x'). | Darjeeling tea is being added, let quantity added be 'x' kg. |
| 3 | Write down the expressions for the new quantities of each component. | New Assam = 35 kg, New Darjeeling = \(14 + x\) kg. |
| 4 | Set up an equation using the new desired ratio. | \(\frac{35}{14 + x} = \frac{2}{1}\). |
| 5 | Solve the equation for 'x'. | Found \(x = 3.5\). |
| 6 | State the final answer. | 3.5 kg of Darjeeling tea must be added. |
Additional Information on Ratios and Mixtures
Ratios are used to compare quantities of the same kind. A ratio \(a : b\) can also be written as a fraction \(\frac{a}{b}\).
Mixture problems often involve ratios to represent the composition of a mixture. When a component is added to or removed from a mixture, the quantities of the components change, and thus the ratio changes.
Understanding how to represent ratios as fractions and solve algebraic equations is key to solving mixture problems like this tea mixture ratio problem.
It's important to ensure that when you set up the ratio equation, the components in the numerator and denominator correspond correctly to the given ratio (e.g., Assam/Darjeeling = 2/1 means Assam quantity divided by Darjeeling quantity equals 2 divided by 1).
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