Which Ratio is Greater: 4:3 or 10:7?

Comparing ratios is a common task in mathematics and can be approached in various ways. In this article, we will explore two methods to determine which of the two ratios, 4:3 and 10:7, is greater. The methods include manipulating the denominators and converting the ratios to fractions.

Method 1: Equalizing Denominators

The first method involves making the denominators of the ratios equal. By doing so, we can easily compare the numerators to determine which ratio is greater.

Starting with the ratio 4:3, we can multiply both the numerator and the denominator by 7 (the second ratio's denominator) to make the denominators equal:

4:3 (4*7):(3*7) 28:21

Similarly, for the ratio 10:7, we multiply both the numerator and the denominator by 3 (the first ratio's denominator) to achieve the same result:

10:7 (10*3):(7*3) 30:21

Now that the denominators are equal, we can compare the numerators. Since 30 is greater than 28, the ratio 10:7 is greater than 4:3. This method is straightforward and involves simple multiplication.

Method 2: Converting to Fractions

Another approach is to convert the ratios into fractions and then compare them. The ratios 4:3 and 10:7 can be written as fractions 4/3 and 10/7, respectively. To compare these fractions, we multiply each by a form of 1 to make their denominators the same:

For 4/3:

4/3 (4/3) * (7/7) 28/21

For 10/7:

10/7 (10/7) * (3/3) 30/21

Again, we have 28/21 and 30/21, and since 30 is greater than 28, we can conclude that 10/7 is greater than 4/3.

Approximate Decimal Form

To further confirm our findings, we can convert the ratios to their decimal forms:

For 4:3:

4:3 ≈ 1.33:1

For 10:7:

10:7 ≈ 1.42:1

Since 1.42 is greater than 1.33, we can confirm that 10:7 is indeed greater than 4:3 in decimal form.

Conclusion

Whether you use the method of equalizing denominators or converting to fractions, the conclusion remains the same: the ratio 10:7 is greater than the ratio 4:3. Both methods provide a clear and simple way to compare ratios and can be easily applied in various mathematical and practical scenarios.

Keywords: ratio comparison, fraction conversion, denominator equalization