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Chapter 11: Problem 10

In Exercises \(4-12,\) solve the proportion. \(\frac{10}{10+z}=\frac{35}{56}\)

Short Answer

Expert verified
z = 6

Step by step solution

01

- Understand the proportion

We are given the proportion \(\frac{10}{10+z}=\frac{35}{56}\). This means that the ratio of 10 to (10+z) is the same as the ratio of 35 to 56.
02

- Cross-multiply to eliminate the fractions

To solve the proportion, we cross-multiply: \(\frac{10}{10+z} = \frac{35}{56}\) leads to \(10 \times 56 = 35 \times (10+z)\).
03

- Simplify the equation

Simplify both sides of the equation obtained from cross-multiplying: \(560 = 35(10+z)\).
04

- Distribute and isolate z

Distribute 35 on the right side: \(560 = 350 + 35z\). To isolate z, subtract 350 from both sides of the equation: \(560 - 350 = 35z\). This simplifies to \(210 = 35z\).
05

- Solve for z

Divide both sides by 35 to solve for z: \(z = \frac{210}{35}\). Simplify this to get \(z = 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cross-multiplication
When solving proportions, cross-multiplication is a powerful tool. A proportion states that two ratios are equal. For example, \(\frac{10}{10+z}=\frac{35}{56}\). This equation means the ratio of 10 to (10+z) is the same as the ratio of 35 to 56.

To cross-multiply, multiply the numerator of each ratio by the denominator of the other ratio. In this case, multiply 10 by 56 and 35 by (10+z). This leads to the equation: \[10 \times 56 = 35 \times (10+z) \]. Now, we can solve the new equation. Cross-multiplication helps transform the proportion into a more straightforward equation.
algebraic equations
Understanding and manipulating algebraic equations is crucial in solving proportions like \(\frac{10}{10+z}=\frac{35}{56}\). After cross-multiplying, we get: \[560 = 35(10+z) \].

Algebraic equations involve variables, constants, and operations. Here, z is the variable to solve for. To isolate z, perform algebraic operations step-by-step:
  • Distribute 35 in \[35(10+z) \]: \[560 = 350 + 35z \].
  • Subtract 350 from both sides: \[560 - 350 = 35z \].
  • Simplify to get: \[210 = 35z \].
By understanding these algebraic operations, you can solve for z.
ratio and proportion
Ratios compare two quantities, while proportions state that two ratios are equal. The given problem, \(\frac{10}{10+z}=\frac{35}{56}\), demonstrates this concept.

In a ratio like \(\frac{10}{10+z}\), 10 and (10+z) are compared. This ratio equals another ratio, \(\frac{35}{56}\)鈥攎eaning they鈥檙e proportional.
  • To solve the proportion, recognize that they maintain the same relationship.
  • Transform initial ratios into an equation using cross-multiplication: \[10 \times 56 = 35 \times (10+z) \].
Proportions help relate quantities in various fields, from recipes to mathematical problems.
simplifying equations
Simplifying equations makes them easier to solve. Start with \[560 = 35(10+z) \].

Simplify by:
  • Distributing: \[560 = 350 + 35z \].
  • Isolating z: \[210 = 35z \].
  • Dividing both sides by 35: \[z = \frac{210}{35} \], which simplifies to z = 6.
Simplification enables solving the equation step-by-step, resulting in the final answer, z = 6. Simplifying equations efficiently translates complex problems into manageable steps.

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Most popular questions from this chapter

Application Romunda's original recipe for her special "cannonball" cookies makes 36 spheres with \(4 \mathrm{cm}\) diameters. She reasons that she can make 36 cannonballs with \(8 \mathrm{cm}\) diameters by doubling the amount of dough. Is she correct? If not, how many 8 cm diameter cannonballs can she make by doubling the recipe?

In Exercises \(4-12,\) solve the proportion. \(\frac{7}{21}=\frac{a}{18}\)

In Exercises \(4-12,\) solve the proportion. \(\frac{10}{b}=\frac{15}{24}\)

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The ratio of the lengths of corresponding diagonals of two similar kites is \(\frac{1}{7} .\) What is the ratio of their areas?
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