Question

If the ratio of integers \(a, b,\) and \(c\) is \(1: 3: 5,\) what is the value of \(a+c ?\) (1) \(c-a=12\) (2) \(b-a=6\)

Short answer

The value of \(a+c\) is 18.

Step-by-step solution

Express Variables in Terms of a Common Multiplier

Given the ratio of integers is 1:3:5, let the common multiplier be represented by the variable k. Thus, express the integers as: \(a = k, \ b = 3k, \ c = 5k\)

Use the First Condition

According to the first condition, \(c - a = 12\). Substitute the expressions from step 1: \[5k - k = 12\] Simplify the equation: \[4k = 12\] Solve for k: \[k = 3\]

Calculate the Values of a and c

Now that we have the value of k, calculate a and c: \(a = k = 3\) \(c = 5k = 5 \times 3 = 15\)

Find the Value of a + c

Add the values of a and c: \(a + c = 3 + 15 = 18\)

Verification Step: Check the Second Condition

To ensure correctness, check the second condition \(b - a = 6\) with the found values: \(b = 3k = 3 \times 3 = 9\) \(b - a = 9 - 3 = 6\). The condition holds true, verifying the result.

Key concepts

ratio and proportion

Understanding the ratio and proportion concept is key to solving many algebra problems. A ratio compares two or more numbers, showing the relative sizes of each number. For example, in the ratio 1:3:5, we know that for every 1 unit of the first number, the second number is 3 units, and the third number is 5 units. By understanding this, you can express the numbers in terms of a common multiplier, helping simplify the problem. When given a ratio, always look to represent each part of the ratio with a variable, multiplying each ratio component by a common multiplier, as shown with k in this exercise.

algebraic equations

In algebra, equations that involve unknown variables can be solved systematically. For this problem, after expressing a, b, and c with the ratio and a common multiplier k, the next step involves creating an algebraic equation with the given conditions. For instance, with the first condition provided, \(c - a = 12\), substitute the expressions \(a = k\) and \(c = 5k\) into the equation: \[5k - k = 12\]. Simplify the left side to get \[4k = 12\], and solve for k by dividing both sides by 4, giving \[k = 3\]. This step highlights the importance of translating word problems into algebraic expressions and solving them using basic algebra rules.

integer properties

When working with integer properties, it's important to remember that integers are whole numbers, both positive and negative, including zero. In this exercise, we're given that a, b, and c are integers. Knowing this, the solution confirms that there are no fractional or decimal parts in calculations. When solving for k (which must also be an integer since a, b, and c are defined by integer ratios), we see that k=3. Then we use k to find a, b, and c as products of k, ensuring that these results also remain integers (a = 3, b = 9, c = 15). Understanding integer properties helps you verify that solutions remain in the correct number set throughout the problem.

common multiplier

A common multiplier simplifies dealing with ratios by letting you express all parts of the ratio using a single variable. Given the ratio 1:3:5, using k as the common multiplier leads to expressions a = k, b = 3k, and c = 5k. By substituting these into the given conditions, you can solve for k. For example, the first condition c-a=12 is transformed into 5k-k=12. Solving this gives 4k=12 and k=3. This common multiplier approach helps break down complex ratio problems into simpler, more manageable equations.

verification steps

Always verify your solution to ensure correctness. For this problem, once you find that a = 3 and c = 15, calculate a + c = 3 + 15 = 18. To verify, use the second condition b - a = 6: knowing b = 9 and a = 3, you check b - a = 9 - 3, which is indeed 6. Since this holds true, it confirms the value of a + c = 18 is correct. Verification steps help ensure your final answers meet all given conditions and highlight any potential calculation errors or misinterpretations.