Question

At Central Park Zoo, the ratio of sea lions to penguins is 4: 11 . If there are 84 more penguins than sea lions, how many sea lions are there? A 24 B 36 C 48 D 72 E 121

Short answer

48

Step-by-step solution

Define Variables

Let the number of sea lions be represented by the variable \( S \) and the number of penguins be represented by the variable \( P \).

Write Down the Ratio Relationships

The ratio of sea lions to penguins is 4 to 11. This can be written as: \[ \frac{S}{P} = \frac{4}{11} \].

Write Down the Difference Relationship

It is given that there are 84 more penguins than sea lions. This can be expressed as: \[ P = S + 84 \].

Express One Variable in Terms of the Other

Since \[ P = S + 84 \], we can substitute \[ P \] in the ratio equation to get: \[ \frac{S}{S + 84} = \frac{4}{11} \].

Solve the Equation for \[ S \]

Cross-multiply to solve for \[ S \]: \[ 11S = 4(S + 84) \]. \ This simplifies to: \[ 11S = 4S + 336 \]. \ Subtract 4S from both sides: \[ 7S = 336 \]. \ Divide both sides by 7: \[ S = 48 \].

Verify Solution

Verify that the solution fits the original conditions. If \[ S = 48 \], then \[ P = 48 + 84 = 132 \]. The ratio \[ \frac{48}{132} = \frac{4}{11} \], which is correct.

Key concepts

Ratios

In this problem, understanding ratios is crucial. A ratio compares two quantities, showing the relationship between them. Here, the ratio of sea lions to penguins is 4:11, meaning for every 4 sea lions, there are 11 penguins. Ratios help set up equations that represent real-world situations. For example, if there are 48 sea lions, the ratio 4:11 lets us determine the corresponding number of penguins.

Variables

Variables allow us to represent unknown quantities with symbols. In this exercise, we use the variable \[ S \], to represent the number of sea lions, and \[ P \], for the number of penguins. Variables make it easier to set up and solve equations. When writing these equations, clearly define each variable to avoid confusion. By stating that \[ S \] is the number of sea lions and \[ P \] is the number of penguins, we're able to create mathematical models of the situation.

Equation Solving

Solving equations involves finding the values of variables that make the equation true. In this problem, we start with two equations: the ratio \[ \frac{S}{P} = \frac{4}{11} \], and the difference \[ P = S + 84 \]. By expressing one variable in terms of the other, we combine these equations into one: \[ \frac{S}{S + 84} = \frac{4}{11} \]. Solving this involves cross-multiplication, simplifying, and isolating the variable. Finally, we ensure the solution fits the original conditions.

Cross-multiplication

Cross-multiplication is a method used to solve equations involving ratios or fractions. It helps clear fractions by multiplying across terms. Starting with \[ \frac{S}{S + 84} = \frac{4}{11} \], we cross-multiply to get \[ 11S = 4(S + 84) \]. Distribute and simplify to find \[ S = 48 \]. This technique is extremely useful for eliminating fractions and solving for unknowns. Always check if cross-multiplication is applicable; it's typically used in proportion problems.