Which of the following is the lowest ratio?
15 ∶ 38
Finding the Lowest Ratio: A Step-by-Step Guide
To find the lowest ratio among the given options, we need to compare their values. The easiest way to compare ratios is to convert them into decimal form. A ratio $a : b$ can be written as a fraction $\frac{a}{b}$. We will calculate the decimal value for each ratio provided in the options.
Calculating Decimal Values for Each Ratio
Let's convert each ratio into a fraction and then calculate its decimal equivalent.
- Ratio 1: 8 : 19
Fraction form: $\frac{8}{19}$
Decimal value: $\frac{8}{19} \approx 0.42105$ - Ratio 2: 15 : 38
Fraction form: $\frac{15}{38}$
Decimal value: $\frac{15}{38} \approx 0.39473$ - Ratio 3: 17 : 38
Fraction form: $\frac{17}{38}$
Decimal value: $\frac{17}{38} \approx 0.44736$ - Ratio 4: 13 : 19
Fraction form: $\frac{13}{19}$
Decimal value: $\frac{13}{19} \approx 0.68421$
Comparing the Decimal Values
Now we have the decimal values for all the ratios. Let's list them out for easy comparison:
| Ratio | Fraction | Approximate Decimal Value |
|---|---|---|
| 8 : 19 | $\frac{8}{19}$ | 0.42105 |
| 15 : 38 | $\frac{15}{38}$ | 0.39473 |
| 17 : 38 | $\frac{17}{38}$ | 0.44736 |
| 13 : 19 | $\frac{13}{19}$ | 0.68421 |
Comparing the decimal values (0.42105, 0.39473, 0.44736, and 0.68421), we can see that 0.39473 is the smallest value.
Identifying the Lowest Ratio
The smallest decimal value, 0.39473, corresponds to the ratio 15 : 38.
Therefore, the lowest ratio among the given options is 15 : 38.
Revision Table: Understanding Ratios
| Concept | Description | Example |
|---|---|---|
| Ratio | A comparison of two quantities by division. Expressed as $a : b$ or $\frac{a}{b}$. | The ratio of apples to oranges is $3 : 2$. |
| Antecedent | The first term in a ratio ($a$ in $a : b$). | In $3 : 2$, the antecedent is 3. |
| Consequent | The second term in a ratio ($b$ in $a : b$). | In $3 : 2$, the consequent is 2. |
| Simplifying Ratios | Dividing both terms by their greatest common divisor (GCD). | The ratio $10 : 15$ simplifies to $2 : 3$ (dividing by GCD of 5). |
Additional Information: Comparing Ratios
There are multiple ways to compare ratios:
- Converting to Decimals: As done in this problem, convert each ratio $\frac{a}{b}$ into its decimal form and compare the decimals. This is often the most straightforward method.
- Converting to Fractions with a Common Denominator: Find a common denominator for all the fractions $\frac{a}{b}$ and $\frac{c}{d}$. Then compare the numerators. The fraction with the smaller numerator (when denominators are equal) is the smaller ratio. For example, to compare $\frac{1}{3}$ and $\frac{2}{5}$, find a common denominator (15). $\frac{1}{3} = \frac{5}{15}$ and $\frac{2}{5} = \frac{6}{15}$. Since $5 < 6$, $\frac{1}{3} < \frac{2}{5}$.
- Cross-Multiplication (for two ratios): To compare $\frac{a}{b}$ and $\frac{c}{d}$, compare $a \times d$ and $b \times c$. If $a \times d < b \times c$, then $\frac{a}{b} < \frac{c}{d}$. If $a \times d > b \times c$, then $\frac{a}{b} > \frac{c}{d}$. If $a \times d = b \times c$, then $\frac{a}{b} = \frac{c}{d}$. This method is efficient for comparing only two ratios at a time.
Choosing the best method depends on the specific ratios given. For multiple ratios, converting to decimals or using a common denominator are generally preferred.
Similar Questions
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Important Questions from Simple Ratios
The ratio of two numbers is 9 : 5. If 8 is added to the larger number and 4 is subtracted from the smaller number, the greater number becomes twice the smaller number. The larger number is:
Divide 500 into two parts such that the ratio of one to the other are in 5 : 3?
The ratio of the number of men and women in a company is 5 : 4. If the number of men and women increase by 16% and 15%, respectively, then what will be the new ratio of men and women ?
The monthly salaries of an officer and a clerk are in the ratio 11 : 4. If the monthly salary of the officer increases by ₹7,000 and that of the clerk by ₹3,000, then the ratio becomes 19 : 7. What was the initial salary (in ₹) of the officer?
If X : Y = 7 : 5 and Y : Z = 7 : 11, then what is the ratio of X : Y : Z?
