The ratio of \(a\) to \(b\) is 3 to \(4,\) and the ratio of \(c\) to \(b\) is 1 to \(2 .\) What is the ratio of \(a\) to \(c ?\) F. 1 to 1 G. 3 to 1 H. 3 to 2 J. 3 to 8 K. 6 to 1

Understanding ratios and proportions is crucial for solving many math problems, including those on the ACT. A ratio compares two quantities, showing how many times one value is contained within the other. It can be written in different forms—using the word 'to', with a colon, or as a fraction, like so: 3 to 4, 3:4, or \( \frac{3}{4} \).

A proportion, on the other hand, is an equation that demonstrates that two ratios are equivalent. If we have a proportion \( \frac{a}{b} = \frac{c}{d} \), we can confidently say that the cross-products are equal, so \( a \times d = b \times c \).

This property is especially useful when working with problems that involve finding an unknown quantity within equivalent ratios. For instance, if we have \( \frac{3}{4} = \frac{a}{b} \) and \( \frac{1}{2} = \frac{c}{b} \), then to find the ratio of \( a \) to \( c \), we may first ensure they have the same denominator and then express each in terms of \( b \).

In the realm of mathematics, especially on tests like the ACT, fluency with fraction operations is essential. The fundamental operations with fractions include simplifying, finding a common denominator, adding, subtracting, multiplying, and dividing.

When dealing with ratios, as seen in the ACT exercise, the ability to manipulate fractions by finding a common denominator assists in comparing or combining ratios. For example, when you have \( \frac{a}{b} \), we want to find an equivalent fraction that has the same denominator as another related fraction, like \( \frac{c}{b} = \frac{1}{2} \).

We achieve this by multiplying both the numerator and the denominator by the needed factor that creates the equivalent fraction. This technique allows us to align the fractions so that the relationship between \( a \) and \( c \) becomes clear. A strong grasp of fraction operations can significantly ease the process of solving ratio problems on the ACT.

Success on the ACT Math section frequently hinges on employing effective strategies. Some key ACT math strategies include:

• Knowing the content: Understand algebra, geometry, ratios, proportions, and other math concepts.
• Managing time: Keep an eye on the clock to ensure you have enough time to address every problem.
• Answering the easier questions first: Quickly move through questions you know how to solve, and come back to the harder ones later.
• Using the process of elimination: Even if you can't find the right answer immediately, eliminate the options that you're certain are incorrect to increase your chances of choosing the correct one.
• Showing your work: Write out the steps to avoid simple mistakes, especially in complex calculations involving fractions or ratios.
• Reviewing your answers: If time allows, go back to double-check your work, particularly on the questions you found challenging.

By mastering these strategies, test-takers can enhance their performance and elevate their scores on the ACT Math section.