Question

Express the ratio $1\frac{1}{2}:3\frac{1}{4}$ using only integers.

Short answer

Question: Convert the mixed numbers 1 1/2 and 3 1/4 into improper fractions, find the ratio between them, and express it using only integers. Answer: 6 : 13

Step-by-step solution

Convert mixed numbers to improper fractions

To convert a mixed number into an improper fraction, first, multiply the whole number by the denominator, then add the numerator, and write the result over the original denominator: $$1\frac{1}{2}=\frac{2\ast 1+1}{2}=\frac{3}{2}$$ $$3\frac{1}{4}=\frac{4\ast 3+1}{4}=\frac{13}{4}$$

Find the Least Common Denominator (LCD)

In order to compare or work with two fractions, we need to have the same denominator for both. The LCD of the denominators 2 and 4 is 4.

Rewrite the fractions with the LCD

Rewrite the fraction $\frac{3}{2}$ with the LCD 4: $$\frac{3}{2}=\frac{3\ast 2}{2\ast 2}=\frac{6}{4}$$ Now we have: $$\frac{6}{4}:\frac{13}{4}$$

Express the ratio using integers

Since both fractions have the same denominator, we can simply express the ratio using the two numerators: $$\frac{6}{4}:\frac{13}{4}=6:13$$ So the ratio expressed using only integers is $6:13$.

Key concepts

Improper Fractions

Improper fractions are a type of fraction where the numerator (the top number) is larger than the denominator (the bottom number). This means that the value of the fraction is greater than one. For example, in the problem provided needing to express the ratio of mixed numbers as integers, converting these mixed numbers into improper fractions is the first step.

To convert a mixed number to an improper fraction, you multiply the whole number part by the denominator and then add the numerator. This value becomes the new numerator, while the denominator remains unchanged. For instance, the mixed number $1\frac{1}{2}$ becomes the improper fraction $\frac{3}{2}$ after applying the conversion process: multiply 1 (whole number) by 2 (denominator), then add the 1 (numerator), resulting in 3, which is placed over the original denominator 2.

Least Common Denominator

The least common denominator (LCD) is the smallest number that can be used as a common denominator for two or more fractions. When comparing or combining fractions with different denominators, converting them to have the LCD is crucial for the operations.

In the exercise, finding an LCD allows us to convert the fractions so they can be directly compared in a ratio. To find the LCD, you determine the smallest number that both denominators can divide into without leaving a remainder. For denominators 2 and 4, the LCD is 4 because it's the smallest multiple they share. Once you've determined the LCD, you can rewrite the fractions by adjusting the numerator to match the new common denominator, making further operations like simplifying a ratio to integers much simpler.

Mixed Numbers

Mixed numbers are numbers that contain both a whole number and a fraction. They represent a value greater than one and are written with a whole number alongside a proper fraction. For example, $3\frac{1}{4}$ is a mixed number, representing three whole units plus a quarter of a unit.

Mixed numbers often need to be converted to improper fractions for calculations, such as when expressing ratios with integers. After conversion, as demonstrated in the exercise, we can handle complex operations more straightforwardly. Following the expression of mixed numbers as improper fractions, they can be more easily compared, added, subtracted, or used in ratios where a clearer, more concise representation is beneficial, especially when the goal is to express the information using only integers.