Question

Translate to an equation and solve. \(\$ 5,710,900\) of the budget at a school comes from the state. If this represents \(65 \%\) of the school's total budget, what is the school's total budget?

Short answer

Answer: The school's total budget is $8,785,230.

Step-by-step solution

Set up a proportion

Let x be the school's total budget. The proportion can be set up as: $$ \frac{65}{100} = \frac{5,710,900}{x} $$

Solve for x

To find the school's total budget (x), cross-multiply and solve for x: $$ 65x = 100(5,710,900) $$

Divide both sides by 65

Divide both sides by 65 to isolate x: $$ x = \frac{100(5,710,900)}{65} $$

Calculate the total budget

Perform the calculation to find the school's total budget: $$ x = \frac{571,090,000}{65} = 8,785,230 $$ The school's total budget is $8,785,230.

Key concepts

Proportion in Mathematics

Understanding proportions is fundamental in mathematics, especially when tackling percentage problems. A proportion is an equation that states that two ratios are equivalent. For example, if half of a pizza has been eaten and half remains, we can express this as a proportion: \( \frac{1}{2} = \frac{eaten}{total} \). In terms of percentages, if we know that \(65\%\) of a school's budget is a certain amount, we're actually dealing with a proportion: \( \frac{65}{100} \) represents the ratio of the state's contribution to the total budget.
By setting up the proportion correctly, we can solve for the unknown quantity which in the context of our exercise is the school's total budget.

Cross-Multiplication

Cross-multiplication is a powerful tool in solving proportions where one of the four numbers is unknown. When we have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying across the equal sign diagonally, giving us \(ad = bc\). This method simplifies the proportion into a solvable equation.
In our exercise, cross-multiplication transforms \( \frac{65}{100} = \frac{5,710,900}{x} \) into the equation, \(65x = 100 \times 5,710,900\), by multiplying the numerator of one ratio by the denominator of the other and vice versa. This step is crucial to isolate the unknown variable, making the equation easier to solve.

Equation Solving

Isolate the Variable

To solve equations like the one we generated through cross-multiplication, the next step is to isolate the variable. This means manipulating the equation so that the unknown variable, \(x\), stands alone on one side and everything else is on the opposite side.

Divide to Solve

The process in our textbook exercise involves dividing both sides of \(65x = 100 \times 5,710,900\) by 65. This division cancels out the 65 on the left side, leaving us with \(x = \frac{100 \times 5,710,900}{65}\), which reveals the total budget upon calculation. The ability to solve for an unknown variable by isolating it is a fundamental skill across many fields of mathematics and beyond.

Mathematical Representation

Mathematical representation includes using numbers, symbols, and equations to model real-world scenarios. This abstraction helps in solving problems that might seem complex at a glance. In our scenario of determining a school's budget from a known percentage, we model the situation with an equation that represents the relationship between the given information and what we need to find.
A clear mathematical representation not only guides us toward the solution but also provides a powerful means of communicating complex information succinctly. It allows us to convert word problems into solvable equations, making problem-solving structured and systematic. This representation is the first and one of the most important steps in attacking any math problem, as seen with our proportion of the school's state budget.