Understanding the Problem: Ratios and Equations

The question asks us to identify which of the given ratio expressions is equivalent to the algebraic equation pq = 3. To solve this, we need to understand how to convert ratios into equivalent algebraic equations and then check which one simplifies to the desired form.

Converting Ratios to Equations

A ratio expressed as a : b is equivalent to the fraction \( \frac{a}{b} \). When we have a proportion (an equality between two ratios), like a : b = c : d, it can be written as \( \frac{a}{b} = \frac{c}{d} \). To remove the fractions, we can cross-multiply, which gives us ad = bc.

Analyzing Each Option

Let's convert each given option from a ratio equation to an algebraic equation and simplify it to see if we get pq = 3.

Option (a) 2p : 3 = 1 : q

Convert the ratio to a fraction equation:

\( \frac{2p}{3} = \frac{1}{q} \)

Cross-multiply:

\( 2p \times q = 3 \times 1 \)

\( 2pq = 3 \)

Divide both sides by 2:

\( pq = \frac{3}{2} \)

This does not match pq = 3.

Option (b) 2 : p = 3q : 1

Convert the ratio to a fraction equation:

\( \frac{2}{p} = \frac{3q}{1} \)

Cross-multiply:

\( 2 \times 1 = p \times 3q \)

\( 2 = 3pq \)

Divide both sides by 3:

\( pq = \frac{2}{3} \)

This does not match pq = 3.

Option (c) 3 : p = q : 2

Convert the ratio to a fraction equation:

\( \frac{3}{p} = \frac{q}{2} \)

Cross-multiply:

\( 3 \times 2 = p \times q \)

\( 6 = pq \)

\( pq = 6 \)

This does not match pq = 3.

Option (d) 3 : p = q : 1

Convert the ratio to a fraction equation:

\( \frac{3}{p} = \frac{q}{1} \)

Cross-multiply:

\( 3 \times 1 = p \times q \)

\( 3 = pq \)

\( pq = 3 \)

This matches the required equation pq = 3.

Conclusion

By converting each ratio option into an algebraic equation, we found that the ratio 3 : p = q : 1 simplifies to the equation pq = 3.

Revision Table: Key Concepts in Ratios and Equations

Additional Information: Proportions and Their Properties

A proportion a : b = c : d means that a, b, c, and d are in proportion. Here, 'a' and 'd' are called the extremes, and 'b' and 'c' are called the means.

The fundamental property of a proportion is that the product of the extremes is equal to the product of the means. For a : b = c : d (or \( \frac{a}{b} = \frac{c}{d} \)), we have:

• Product of extremes = \( a \times d \)
• Product of means = \( b \times c \)
• Thus, \( ad = bc \)

This property is exactly what we used when cross-multiplying to convert the ratio equations into simpler algebraic forms like pq = 3. Understanding this property is key to solving problems involving ratios and proportions.

Ratios and proportions are used in various fields, including scaling recipes, map reading, mixing chemicals, and calculating speeds and distances. Being able to manipulate ratio equations is a fundamental skill in mathematics and many practical applications.