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A **proportion** is an equation of the form $$\frac{a}{b} = \frac{c}{d}$$, where $$b \neq 0$$ and $$d \neq 0$$, asserting that two rational expressions are equal. This equation is read as "$$a$$ is to $$b$$, as $$c$$ is to $$d$$." For instance, $$\frac{1}{2} = \frac{4}{8}$$ is a proportion — read "1 is to 2 as 4 is to 8" — because both fractions represent the same value. Proportions frequently appear in applications that involve scaling a known ratio to a larger or smaller context; when an unknown quantity occupies any of the four positions, it can be found by solving the equation. Because a proportion is a rational equation, it is solved by multiplying both sides by the LCD to clear the fractions and then solving the resulting equation. As with all rational expressions, any variable values that would make either denominator equal zero must be excluded.

Proportion

Solve the proportion $$\frac{x}{63} = \frac{4}{7}$$.

**Step 1 — Multiply both sides by the LCD.** The denominators are $$63$$ and $$7$$. Since $$63 = 7 \cdot 9$$, the LCD is $$63$$. Multiply both sides by $$63$$:

$$63 \cdot \frac{x}{63} = 63 \cdot \frac{4}{7}$$

**Step 2 — Simplify.** On the left, $$63$$ cancels with the denominator $$63$$, leaving $$x$$. On the right, $$\frac{63}{7} = 9$$, so the product is $$9 \cdot 4 = 36$$:

$$x = 36$$

**Step 3 — Check.** Substitute $$x = 36$$ back into the original proportion:

$$\frac{36}{63} = \frac{4 \cdot 9}{7 \cdot 9} = \frac{4}{7}$$ ✓

The solution is $$x = 36$$. This example illustrates the basic technique for solving a proportion when the variable appears in a numerator: multiply both sides by the LCD to clear fractions, divide out common factors, and verify by substituting the result into the original equation.

Solving $$\frac{x}{63} = \frac{4}{7}$$

Solve the proportion $$\frac{144}{a} = \frac{9}{4}$$.

Because the variable $$a$$ appears in a denominator, the value $$a = 0$$ must be excluded — it would make the left-hand expression undefined.

**Step 1 — Multiply both sides by the LCD.** The denominators are $$a$$ and $$4$$. The LCD is $$4a$$. Multiply both sides by $$4a$$:

$$4a \cdot \frac{144}{a} = 4a \cdot \frac{9}{4}$$

**Step 2 — Remove common factors on each side.** On the left, $$a$$ cancels with the denominator, giving $$4 \cdot 144 = 576$$. On the right, $$4$$ cancels with the denominator, giving $$9a$$:

$$576 = 9a$$

**Step 3 — Divide both sides by $$9$$:**

$$\frac{576}{9} = \frac{9a}{9}$$

$$64 = a$$

**Step 4 — Check.** Substitute $$a = 64$$ into the original proportion:

$$\frac{144}{64} = \frac{9 \cdot 16}{4 \cdot 16} = \frac{9}{4}$$ ✓

The solution is $$a = 64$$. Unlike proportions where the variable sits in a numerator, here the variable appears in a denominator, so the LCD contains the variable ($$4a$$). After clearing fractions, the equation reduces to a simple linear equation that is solved by dividing both sides by the coefficient.

Solving $$\frac{144}{a} = \frac{9}{4}$$

Apply the proportion-solving strategy to a real-world medical dosage problem.

**Problem:** Pediatricians prescribe $$5$$ milliliters (ml) of acetaminophen for every $$25$$ pounds of a child's weight. If Zoe weighs $$80$$ pounds, how many milliliters of acetaminophen will her doctor prescribe?

1. **Identify** what to find and assign a variable: How many ml of acetaminophen should be prescribed? Let $$a$$ = ml of acetaminophen.
2. **Translate** into a proportion, keeping units consistent on each side — milliliters over pounds equals milliliters over pounds:

$$\frac{5}{25} = \frac{a}{80}$$

3. **Solve** by multiplying both sides by the LCD, which is $$400$$:

$$400 \cdot \frac{5}{25} = 400 \cdot \frac{a}{80}$$

Remove common factors on each side. On the left, $$\frac{400}{25} = 16$$, so the left side becomes $$16 \cdot 5$$. On the right, $$\frac{400}{80} = 5$$, so the right side becomes $$5a$$:

$$16 \cdot 5 = 5a$$

Divide both sides by $$5$$:

$$16 = a$$

4. **Check reasonableness:** Since $$80$$ is approximately $$3$$ times $$25$$, the dosage should be roughly $$3$$ times $$5$$, which is $$15$$. The answer of $$16$$ ml is close and makes sense.
5. **Verify** by substituting $$a = 16$$ back into the original proportion:

$$\frac{5}{25} = \frac{16}{80}$$

$$\frac{1}{5} = \frac{1}{5}$$ ✓

**Answer:** The pediatrician would prescribe $$16$$ ml of acetaminophen to Zoe.

This example demonstrates a key technique in proportion word problems: aligning the units so that both ratios compare the same quantities in the same order (ml over pounds on both sides). The LCD method then transforms the proportion into a simple linear equation.

Solving a Proportion Application: Medication Dosage by Child's Weight

Apply the proportion-solving strategy to determine the calorie content of a differently sized beverage.

**Problem:** A 16-ounce iced caramel macchiato has 230 calories. How many calories are there in a 24-ounce iced caramel macchiato?

1. **Identify** what to find and assign a variable: How many calories are in the 24-ounce drink? Let $$c$$ = calories in 24 ounces.
2. **Translate** into a proportion, keeping units consistent — calories over ounces equals calories over ounces:

$$\frac{230}{16} = \frac{c}{24}$$

3. **Solve** by multiplying both sides by the LCD, which is $$48$$:

$$48 \cdot \frac{230}{16} = 48 \cdot \frac{c}{24}$$

Remove common factors on each side. On the left, $$\frac{48}{16} = 3$$, so the left side becomes $$3 \cdot 230 = 690$$. On the right, $$\frac{48}{24} = 2$$, so the right side becomes $$2c$$:

$$690 = 2c$$

Divide both sides by $$2$$:

$$345 = c$$

4. **Check reasonableness:** 345 calories for a 24-ounce drink is more than 230 calories for a 16-ounce drink, but not excessively so — this is consistent with a proportional increase in size.
5. **Verify** by substituting $$c = 345$$ back into the original proportion:

$$\frac{230}{16} = \frac{345}{24}$$

$$\frac{115}{8} = \frac{115}{8}$$ ✓

**Answer:** There are 345 calories in a 24-ounce iced caramel macchiato.

Like the medication dosage problem, this example reinforces the importance of setting up the proportion with matching units on each side (calories over ounces). The LCD of 48 is found from the denominators 16 and 24, and after clearing fractions the equation reduces to a straightforward division.

Solving a Proportion Application: Calories in a Larger Beverage

Apply the proportion-solving strategy to a currency exchange problem.

**Problem:** Josiah went to Mexico for spring break and changed \$325 into Mexican pesos. At that time, the exchange rate was \$1 US equal to 12.54 Mexican pesos. How many Mexican pesos did he get for his trip?

1. **Identify** what to find and assign a variable: How many Mexican pesos did Josiah get? Let $$p$$ = the number of Mexican pesos.
2. **Translate** into a proportion, keeping units consistent — dollars over pesos equals dollars over pesos:

$$\frac{1}{12.54} = \frac{325}{p}$$

3. **Solve** by multiplying both sides by the LCD, $$12.54p$$:

$$12.54p \cdot \frac{1}{12.54} = 12.54p \cdot \frac{325}{p}$$

Remove common factors on each side. On the left, $$12.54$$ cancels, leaving $$p$$. On the right, $$p$$ cancels, leaving $$12.54 \cdot 325$$:

$$p = 4075.5$$

4. **Check reasonableness:** \$100 would be 1,254 pesos. Since \$325 is a little more than 3 times \$100, the answer of 4,075.5 pesos makes sense (roughly $$3 \times 1{,}254 = 3{,}762$$, and 4,075.5 is close to that estimate).
5. **Verify** by substituting $$p = 4075.5$$ back into the original proportion:

$$\frac{1}{12.54} = \frac{325}{4075.5}$$

$$0.07974 \approx 0.07974$$ ✓

**Answer:** Josiah got 4,075.5 Mexican pesos for his spring break trip.

This example demonstrates that proportion problems involving currency exchange follow the same procedure as other proportion applications: align matching units on each side of the proportion (dollars over pesos = dollars over pesos), multiply both sides by the LCD to clear the fractions, and verify the result. The number of pesos is **proportional** to the number of dollars — when two quantities are related by a proportion, they are said to be proportional.

Solving a Proportion Application: Currency Exchange from Dollars to Pesos

In workplace applications such as scaling a blueprint or adjusting a budget, what is the specific term for an equation that states two ratios are equal?

In a professional setting, when a team lead needs to scale a known ratio to a larger context by creating an equation that asserts two ratios are equal, this equation is known as a ____.

In a professional environment, understanding how to compare and scale values is essential. Match each term below with the scenario or description that best defines its role in workplace mathematics based on the provided concept of proportions.

In a proportion used for scaling professional ratios, such as $\frac{a}{b} = \frac{c}{d}$, the values in the denominators ($b$ and $d$) must not be equal to zero.

In many professional fields, you must scale a known ratio to a larger or smaller context by using a 'proportion.' Provide the formal mathematical definition of a proportion and write its standard algebraic form, including the necessary restrictions for the variables located in the denominators.

The Mathematical Structure of Proportions in Scaling

Suppose you are a logistics manager who needs to scale a fuel-to-mileage ratio for a larger vehicle fleet. To find an unknown value using a proportion, arrange the following steps in the correct order to solve the equation using the method of clearing fractions.

As part of a professional development initiative, you are contributing to a glossary of mathematical terms used in project management. Your task is to write a concise entry for the term **proportion** to help new employees understand how ratios are used in scaling project costs.

To ensure the entry is complete, your response must:
1. Provide the formal definition of a 'proportion' as it relates to rational expressions.
2. Show the general algebraic form used to represent a proportion.
3. Explain how a proportion (such as 1/2 = 4/8) is read aloud using the standard 'is to' phrasing.
4. Identify the specific mathematical restriction that must be applied to the values in the denominators of a proportion.

Defining Proportions for Workplace Documentation

Based on the logistics coordinator's scenario, identify the specific mathematical name for the equation used to scale the fuel ratio and state the formal requirement regarding the denominators for this equation to be valid.

Fuel Efficiency Scaling in Logistics

In a manufacturing facility, a quality control technician uses the proportion $\frac{4}{25} = \frac{16}{100}$ to demonstrate that the ratio of defective parts remains consistent as production scales. According to the formal definition of a proportion, how should the technician read this equation during a shift report?

A logistics coordinator is comparing the fuel efficiency of two different delivery trucks by documenting their miles-per-gallon ratios. If the coordinator records these rates as a **proportion**, what does this mathematical equation specifically assert about the two ratios?

An alternative method to clearing fractions by multiplying by the least common denominator in a proportion of the form $$\frac{a}{b} = \frac{c}{d}$$ is to cross-multiply. This process involves multiplying the numerator of each fraction by the denominator of the other, setting the two products equal to each other to obtain $$ad = bc$$. Both cross-multiplication and multiplying by the least common denominator yield the exact same fraction-free equation.

Solving Proportions using Cross-Multiplication

To find the solution to the proportion $$\frac{n}{n+14} = \frac{5}{7}$$, first eliminate the fractions by multiplying both sides by the least common denominator, $$7(n+14)$$. This step simplifies the equation to $$7n = 5(n+14)$$. Next, distribute the $$5$$ to yield $$7n = 5n + 70$$. Subtract $$5n$$ from both sides to isolate the variable terms, resulting in $$2n = 70$$. Dividing both sides by $$2$$ provides the solution $$n = 35$$. The result can be verified by substituting $$35$$ back into the original proportion, yielding $$\frac{35}{49} = \frac{5}{7}$$, which is a true statement.

Solving the Proportion $$\frac{n}{n+14} = \frac{5}{7}$$

When formulating a proportion to model a word problem, it is essential to align the units properly across the equation. The units in the numerators of both fractions must correspond to each other, and similarly, the units in the denominators must match. Maintaining this consistency ensures that the resulting rational equation accurately reflects the problem's real-world conditions before algebraic solving techniques are applied.

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