Question

The ratio of \(x\) to \(y\) is \(1: 4 .\) If the value of \(x\) were increased by 1 without changing the value of \(y\), the ratio of \(x\) to \(y\) would become \(1: 3 .\) If \(z=x+2 y,\) what is the value of \(z ?\)

Short answer

The value of \(z\) is 27.

Step-by-step solution

Write the given ratios as equations

The ratio of \(x\) to \(y\) is given as \(1: 4\), which means \(\frac{x}{y} = \frac{1}{4}\). So we can write it as: \(x = \frac{1}{4}y\).

Express the new ratio

If \(x\) is increased by 1, the new ratio given is \(1: 3\). So, \(\frac{x+1}{y} = \frac{1}{3}\).

Set up the new equation

Rewrite the second ratio equation as: \(x + 1 = \frac{1}{3}y\).

Substitute the first equation into the second equation

Substitute \(x = \frac{1}{4}y\) into \(x + 1 = \frac{1}{3}y\): \(\frac{1}{4}y + 1 = \frac{1}{3}y\).

Solve for \(y\)

Multiply all terms by 12 to clear the fractions: \(3y + 12 = 4y\). Simplify: \(12 = y\).

Find \(x\)

Use \(x = \frac{1}{4}y\) with \(y = 12\): \(x = \frac{1}{4} \times 12 = 3\).

Calculate \(z\)

Given \(z = x + 2y\), substitute \(x = 3\) and \(y = 12\): \(z = 3 + 2 \times 12 = 3 + 24 = 27\).

Key concepts

Understanding Ratios in Algebra

In algebra, ratios are a way to compare two quantities by division. For instance, the ratio of two numbers, say \(a\) and \(b\), can be written as \(a:b\) or \(\frac{a}{b}\). This indicates how many times one number contains another. In this exercise, the ratio of \(x\) to \(y\) is given as 1:4, meaning if \(y\) is 4 times a unit, \(x\) is 1 unit in comparison.
To convert the ratio into an algebraic equation, set \(\frac{x}{y} = \frac{1}{4}\) which reduces to \(x = \frac{1}{4}y\). Using ratios in equations helps us understand the relationship between variables in a precise manner.

Applying the Substitution Method

The substitution method is a powerful tool for solving algebraic equations, especially when you have multiple equations involving the same variables. Here, we initially find that \(x = \frac{1}{4}y\). When we're told that increasing \(x\) by 1 changes the ratio to 1:3, it gives us another equation: \(\frac{x+1}{y} = \frac{1}{3}\).
By rewriting this, we get \(x + 1 = \frac{1}{3} y\). To use the substitution method, we substitute \(x = \frac{1}{4} y\) from the first equation into the second equation. This substitution simplifies our equations and allows us to isolate and solve for one variable at a time.
For instance, substituting gives us \(\frac{1}{4}y + 1 = \frac{1}{3}y\). Solving these equations step-by-step narrows down the values of \(y\) and subsequently helps in finding \(x\).

Solving Algebraic Equations

Algebraic equations are mathematical statements with an equality symbol connecting two expressions. To solve these equations, especially those involving ratios and multiple variables, it's important to follow systematic steps like isolating variables and performing arithmetic operations correctly.
From the substitution, we arrive at \(\frac{1}{4} y + 1 = \frac{1}{3} y\). Clearing the fractions by multiplying all terms by 12 gives: \(3y + 12 = 4y\). This step eliminates fractions making it easier to solve for \(y\). Simplifying further, we solve \(3y + 12 = 4y\) to find \(y = 12\).
Next, substituting \(y = 12\) back into \(x = \frac{1}{4} y\) helps to find \(x\). Finally, using \(z = x + 2y\), substituting \(x = 3\) and \(y = 12\) gives \(z = 27\). This methodical approach is crucial for correct solutions in algebra.