Finding the Savings Ratio of Playback Singers

The problem provides us with the income ratios and expenditure ratios of three playback singers: Aswath, Sunita, and Ramdin. We are also given specific information about Aswath's savings as a percentage of his income. Our goal is to determine the ratio of their individual savings.

Setting up the Ratios with Variables

Let's represent the incomes and expenditures using variables based on the given ratios:

• The income ratio of Aswath, Sunita, and Ramdin is \(12 : 9 : 7\). Let their incomes be \(12x\), \(9x\), and \(7x\) respectively, where \(x\) is a common positive multiplier.
• The expenditure ratio of Aswath, Sunita, and Ramdin is \(15 : 9 : 8\). Let their expenditures be \(15y\), \(9y\), and \(8y\) respectively, where \(y\) is another common positive multiplier.

Using Aswath's Saving Information

We know that Saving = Income - Expenditure. For Aswath, we are given that he saves 25% of his income.

• Aswath's Income = \(12x\)
• Aswath's Expenditure = \(15y\)
• Aswath's Saving = 25% of his Income = \(0.25 \times 12x = \frac{1}{4} \times 12x = 3x\).

Using the Saving = Income - Expenditure formula for Aswath:

\(3x = 12x - 15y\)

Now, let's rearrange this equation to find a relationship between \(x\) and \(y\):

\(15y = 12x - 3x\)

\(15y = 9x\)

Dividing both sides by 3, we get:

\(5y = 3x\)

From this relationship, we can express \(y\) in terms of \(x\):

\(y = \frac{3}{5}x\)

Calculating Savings for Each Singer

Now we can calculate the saving for each singer using the formula Saving = Income - Expenditure, substituting the expression for \(y\).

• Aswath's Saving:
  \(S_{A} = 12x - 15y\) Substitute \(y = \frac{3}{5}x\):
  \(S_{A} = 12x - 15\left(\frac{3}{5}x\right) = 12x - \frac{45}{5}x = 12x - 9x = 3x\) (This matches the 25% of income we calculated earlier, which confirms our relationship between \(x\) and \(y\)).
• Sunita's Saving:
  \(S_{S} = 9x - 9y\) Substitute \(y = \frac{3}{5}x\):
  \(S_{S} = 9x - 9\left(\frac{3}{5}x\right) = 9x - \frac{27}{5}x\) To subtract, find a common denominator:
  \(S_{S} = \frac{9 \times 5}{5}x - \frac{27}{5}x = \frac{45}{5}x - \frac{27}{5}x = \frac{45 - 27}{5}x = \frac{18}{5}x\)
• Ramdin's Saving:
  \(S_{R} = 7x - 8y\) Substitute \(y = \frac{3}{5}x\):
  \(S_{R} = 7x - 8\left(\frac{3}{5}x\right) = 7x - \frac{24}{5}x\) To subtract, find a common denominator:
  \(S_{R} = \frac{7 \times 5}{5}x - \frac{24}{5}x = \frac{35}{5}x - \frac{24}{5}x = \frac{35 - 24}{5}x = \frac{11}{5}x\)

Finding the Ratio of Savings

The ratio of the savings of Aswath, Sunita, and Ramdin is \(S_{A} : S_{S} : S_{R}\).

\(S_{A} : S_{S} : S_{R} = 3x : \frac{18}{5}x : \frac{11}{5}x\)

Since \(x\) is a common positive multiplier for income, we can cancel it out from the ratio:

\(3 : \frac{18}{5} : \frac{11}{5}\)

To get rid of the fractions, multiply the entire ratio by the least common multiple of the denominators, which is 5:

\(3 \times 5 : \frac{18}{5} \times 5 : \frac{11}{5} \times 5\)

\(15 : 18 : 11\)

Thus, the ratio of the savings of Aswath, Sunita, and Ramdin is \(15 : 18 : 11\).

Summary of Savings

Ratio of Savings: \(3x : \frac{18}{5}x : \frac{11}{5}x\)

Multiply by 5: \(15x : 18x : 11x\)

Simplify by dividing by \(x\): \(15 : 18 : 11\)

This ratio matches one of the given options.

Revision Table: Key Concepts for Ratio Problems

Additional Information: Ratios and Proportions

Understanding ratios and how to work with them is fundamental in many quantitative problems. A ratio expresses the relative sizes of two or more values.

• When dealing with ratios like \(a:b\), it means \(a/b\). If you have a ratio \(a:b:c\), it means the quantities are in proportion \(ak, bk, ck\) for some non-zero constant \(k\).
• To compare or combine ratios, it is often helpful to introduce variables, as we did with \(x\) and \(y\) for income and expenditure.
• If you have ratios involving fractions (e.g., \(1/2 : 1/3\)), you can simplify them by multiplying by the least common multiple of the denominators to get whole numbers (e.g., multiply by 6 to get \(3:2\)). In this problem, we multiplied by 5 to remove fractions in the savings ratio.
• Saving is the part of the income that is not spent. If income is I, expenditure is E, and saving is S, then \(I = E + S\) or \(S = I - E\).
• Percentages can be easily converted to decimals or fractions for calculations. 25% is \(0.25\) or \(1/4\).

This type of problem requires combining the concepts of ratios, percentages, and basic algebra to solve for an unknown ratio.