Chapter 8.1 Rate - Ratio Simplification Techniques

Fundamentals of Ratio Simplification

The simplification of ratios is a foundational skill in mathematics, specifically within the study of rates and proportions. A ratio represents the relative sizes of two or more values. To simplify a ratio, one must express the terms in their smallest possible integer form while maintaining the same relationship between them. This process is analogous to simplifying fractions; however, ratios can involve more than two terms and frequently include different units of measurement, decimals, or mixed numbers. The primary goal of simplification is to reach a state where all terms are whole numbers and their greatest common divisor (GCD) is $1$, meaning the terms are coprime.

Methodologies for Handling Fractional and Mixed Number Ratios

When a ratio consists of fractions or mixed numbers, the standard procedure involves several discrete steps to ensure accuracy. First, any mixed numbers must be converted into improper fractions. For example, if given the term $2 \frac{1}{4}$, it is converted to $\frac{9}{4}$ by multiplying the whole number by the denominator and adding the numerator. Once all terms are in fractional form, the next step is to find the Least Common Multiple (LCM) of all the denominators involved. Multiplying every term in the ratio by this LCM effectively eliminates the denominators, leaving only the numerators as integers.

Consider the ratio $2 \frac{1}{4} : 1 \frac{3}{5}$. The conversion to improper fractions yields $\frac{9}{4} : \frac{8}{5}$. To remove the denominators $4$ and $5$, we identify the LCM as $20$. Multiplying both sides by $20$ results in $( \frac{9}{4} \times 20 ) : ( \frac{8}{5} \times 20 )$. This simplifies to $( 9 \times 5 ) : ( 8 \times 4 )$, which equals $45 : 32$. Because $45$ and $32$ share no common factors other than $1$, the ratio is in its simplest form. Similarly, for the ratio $\frac{3}{4} : 1 \frac{2}{3}$, the improper fractions are $\frac{3}{4} : \frac{5}{3}$. Using the LCM of $12$, the calculation becomes $( \frac{3}{4} \times 12 ) : ( \frac{5}{3} \times 12 )$, resulting in $9 : 20$.

Procedural Approach to Decimal and Multi-Term Ratios

Simplifying ratios containing decimals requires a technique to shift the decimal point until all values become integers. This is achieved by multiplying all terms by the same power of $10$ (such as $10$, $100$, or $1000$), depending on the term with the most decimal places. For the ratio $0.2 : 0.28$, the term $0.28$ has two decimal places, necessitating a multiplication of all terms by $100$. This transforms the ratio into $20 : 28$. Once in integer form, the ratio can be simplified further by dividing by the common factor of $4$, resulting in a final answer of $5 : 7$.

Multi-term ratios follow the same logic but require the operation to be applied to every component simultaneously. In the case of $3 \frac{2}{5} : 0.4 : 8$, the first step is to convert all terms to a consistent format, typically decimals or fractions. Converting to decimals gives $3.4 : 0.4 : 8$. To eliminate the decimals, we multiply by $10$ to obtain $34 : 4 : 80$. Each of these even numbers can then be divided by $2$, yielding the simplified final ratio of $17 : 2 : 40$. Another example of a multi-term ratio is $\frac{1}{4} : 2 : \frac{1}{5}$. To simplify this, the LCM of the denominators $4$ and $5$ (which is $20$) is used to multiply each term: $\frac{1}{4} \times 20 : 2 \times 20 : \frac{1}{5} \times 20$. This results in the simplified integer ratio of $5 : 40 : 4$.

Harmonization of Units in Ratios

A critical rule in ratio simplification is that all terms must represent the same units before they can be compared or simplified numerically. If the units differ, such as hours versus minutes or dollars versus cents, a conversion must take place first. It is generally most efficient to convert the larger unit into the smaller unit to avoid introducing additional fractions or decimals.

For the calculation of $1 \frac{2}{3} \text{ h} : 48 \text{ min}$, we first acknowledge the conversion factor of $1 \text{ h} = 60 \text{ min}$. The hours are converted as follows: $1 \frac{2}{3} \times 60 = \frac{5}{3} \times 60 = 100 \text{ min}$. The ratio is then written as $100 \text{ min} : 48 \text{ min}$. Dividing both terms by the common factor of $4$ results in $25 : 12$. Similarly, for currency ratios such as 90 \cent : \$2.70, we use the fact that \$1 = 100 \cent. Converting the dollars gives us 2.70 \times 100 = 270 \cent. The resulting ratio, $90 : 270$, can be simplified by dividing by $90$, which yields the final result of $1 : 3$.

Summary of Worked Problem Sets

The following is a comprehensive list of simplified ratios based on the provided material, intended for rigorous study and verification of methods. For each problem, the original ratio is presented followed by the necessary transformation steps and the final simplified integer ratio:

1. Ratio: $2 \frac{1}{4} : 1 \frac{3}{5}$ Process: $\frac{9}{4} : \frac{8}{5} \rightarrow \frac{45}{20} : \frac{32}{20}$ Result: $45 : 32$

2. Ratio: $\frac{3}{4} : 1 \frac{2}{3}$ Process: $\frac{3}{4} : \frac{5}{3} \rightarrow \frac{9}{12} : \frac{20}{12}$ Result: $9 : 20$

3. Ratio: $0.2 : 0.28$ Process: $20 : 28 \rightarrow \frac{20}{4} : \frac{28}{4}$ Result: $5 : 7$

4. Ratio: $3 \frac{2}{5} : 0.4 : 8$ Process: $3.4 : 0.4 : 8 \rightarrow 34 : 4 : 80 \rightarrow 17 : 2 : 40$ Result: $17 : 2 : 40$

5. Ratio: $1 \frac{2}{3} \text{ h} : 48 \text{ min}$ Process: $100 \text{ min} : 48 \text{ min} \rightarrow \frac{100}{4} : \frac{48}{4}$ Result: $25 : 12$

6. Ratio: 90 \cent : \$2.70 Process: 90 \cent : 270 \cent \rightarrow \frac{90}{90} : \frac{270}{90} Result: $1 : 3$

7. Ratio: $\frac{1}{4} : 2 : \frac{1}{5}$ Process: $( \frac{1}{4} \times 20 ) : ( 2 \times 20 ) : ( \frac{1}{5} \times 20 )$ Result: $5 : 40 : 4$