Calculating Percentage Difference in Election Votes

The problem provides election data for the year 2018, specifically focusing on the ratios of different types of votes. We are given two key ratios:

• The ratio of the number of votes polled to the number of valid votes is 5 ∶ 4.
• The ratio of valid votes of male and females is 5 ∶ 3.

We need to find out the percentage by which the number of valid votes by females in 2018 is less than the number of votes polled in the same year.

Step-by-Step Calculation

Let's denote:

• Total votes polled in 2018 as $V_{polled}$.
• Total valid votes in 2018 as $V_{valid}$.
• Valid male votes in 2018 as $V_{male\_valid}$.
• Valid female votes in 2018 as $V_{female\_valid}$.

Relating Total Votes and Valid Votes

We are given that the ratio of votes polled to valid votes is 5 ∶ 4.

This can be written as:

$$\frac{V_{polled}}{V_{valid}} = \frac{5}{4}$$

From this, we can express the number of valid votes in terms of the votes polled:

$$V_{valid} = \frac{4}{5} \times V_{polled}$$

Relating Valid Votes of Males and Females

We are given that the ratio of valid votes of male to females is 5 ∶ 3.

The total number of parts in this ratio is $5 + 3 = 8$.

The valid female votes ($V_{female\_valid}$) represent 3 parts out of the total 8 parts of the valid votes. Therefore:

$$V_{female\_valid} = \frac{3}{8} \times V_{valid}$$

Finding Valid Female Votes in terms of Total Votes Polled

Now we can substitute the expression for $V_{valid}$ into the equation for $V_{female\_valid}$:

$$V_{female\_valid} = \frac{3}{8} \times \left(\frac{4}{5} \times V_{polled}\right)$$

Let's simplify this expression:

$$V_{female\_valid} = \frac{3 \times 4}{8 \times 5} \times V_{polled} = \frac{12}{40} \times V_{polled}$$

Reducing the fraction $\frac{12}{40}$:

$$\frac{12}{40} = \frac{3 \times 4}{10 \times 4} = \frac{3}{10}$$

So, the number of valid female votes is:

$$V_{female\_valid} = \frac{3}{10} \times V_{polled}$$

This means valid female votes are 3/10 of the total votes polled.

Calculating the Percentage Less

We want to find how much less the valid female votes are compared to the total votes polled, expressed as a percentage of the total votes polled.

The difference is $V_{polled} - V_{female\_valid}$.

The percentage less is given by the formula:

$$\text{Percentage less} = \frac{V_{polled} - V_{female\_valid}}{V_{polled}} \times 100\%$$

Substitute the expression for $V_{female\_valid}$:

$$\text{Percentage less} = \frac{V_{polled} - \left(\frac{3}{10} \times V_{polled}\right)}{V_{polled}} \times 100\%$$

Factor out $V_{polled}$ from the numerator:

$$\text{Percentage less} = \frac{V_{polled} \times \left(1 - \frac{3}{10}\right)}{V_{polled}} \times 100\%$$

Cancel out $V_{polled}$ (assuming $V_{polled} \neq 0$):

$$\text{Percentage less} = \left(1 - \frac{3}{10}\right) \times 100\%$$

Calculate the difference in the parenthesis:

$$1 - \frac{3}{10} = \frac{10}{10} - \frac{3}{10} = \frac{10 - 3}{10} = \frac{7}{10}$$

Finally, calculate the percentage:

$$\text{Percentage less} = \frac{7}{10} \times 100\% = 0.7 \times 100\% = 70\%$$

So, the number of valid votes by females in 2018 is 70% less than the number of votes polled in the same year.

Difference = $V_{polled} - V_{female\_valid} = V_{polled} - \frac{3}{10} V_{polled} = \frac{7}{10} V_{polled}$.

Percentage Less = $\frac{\text{Difference}}{\text{Votes Polled}} \times 100\% = \frac{\frac{7}{10} V_{polled}}{V_{polled}} \times 100\% = \frac{7}{10} \times 100\% = 70\%$.

Conclusion

The number of valid votes by females in 2018 is 70% less than the number of votes polled in the same year.

Revision Table: Election Data Ratios

Additional Information: Understanding Ratios and Percentages

This problem involves working with ratios and converting them into fractions to find proportional parts of a whole. Understanding how to combine ratios and calculate percentage differences is crucial for data interpretation questions.

• Ratio: A ratio compares two or more quantities. A ratio $a : b$ means that for every $a$ units of the first quantity, there are $b$ units of the second quantity.
• Converting Ratio to Fraction: If a whole is divided into parts in the ratio $a : b$, the total number of parts is $a+b$. The fraction of the whole that the first quantity represents is $\frac{a}{a+b}$, and the fraction for the second quantity is $\frac{b}{a+b}$.
• Percentage Difference: The percentage difference between two values A and B, relative to A, is calculated as $\frac{|A - B|}{A} \times 100\%$. In this problem, we needed to find how much less B is than A, so the difference is A - B, and the percentage less is $\frac{A - B}{A} \times 100\%$.

By setting up the relationships between the different categories of votes using the given ratios, we could express the quantity of valid female votes as a fraction of the total votes polled, allowing us to calculate the required percentage difference.