Master the technique of cross-multiplication to solve for unknown values in proportions. Apply these skills to solve word problems involving direct variation, scaling on maps, and adjusting recipe ingredients.
What is the principle of cross-multiplication for solving a proportion \(\frac{a}{b} = \frac{c}{d}\)?
To solve the proportion, you multiply the numerator of each ratio by the denominator of the other, setting them equal. This gives the equation \(a \times d = b \times c\). You can then solve for any unknown variable.
Solve for \(x\) in the proportion: \(\frac{5}{8} = \frac{x}{32}\)
Using cross-multiplication: \(5 \times 32 = 8 \times x\). This gives \(160 = 8x\). Divide both sides by \(8\) to get \(x = 20\).
A map has a scale of \(1\) inch : \(25\) miles. If the distance between two cities on the map is \(4.5\) inches, what is the actual distance?
Set up a proportion: \(\frac{1 \text{ inch}}{25 \text{ miles}} = \frac{4.5 \text{ inches}}{x \text{ miles}}\). Cross-multiply to get \(1 \cdot x = 25 \cdot 4.5\). Solving gives \(x = 112.5\) miles.
What does it mean for two quantities, \(y\) and \(x\), to be in direct proportion (or vary directly)?
It means that as \(x\) increases, \(y\) increases at the same rate, and as \(x\) decreases, \(y\) decreases at the same rate. Their ratio is constant.
What is the general equation for direct variation, and what does the variable \(k\) represent?
The equation is \(y = kx\). The variable \(k\) is called the constant of proportionality. It represents the constant ratio of \(y\) to \(x\) (i.e., \(k = \frac{y}{x}\)).
If \(y\) varies directly with \(x\), and \(y = 12\) when \(x = 3\), what is the constant of proportionality \(k\)?
Using the formula \(k = \frac{y}{x}\), substitute the given values: \(k = \frac{12}{3} = 4\). The equation for this relationship is \(y = 4x\).
If \(y\) varies directly with \(x\), and \(y = 10\) when \(x = 2\), what is the value of \(y\) when \(x = 7\)?
First, find the constant of proportionality: \(k = \frac{y}{x} = \frac{10}{2} = 5\). The equation is \(y = 5x\). Now, substitute \(x = 7\) into the equation: \(y = 5(7) = 35\).
A recipe for \(12\) cookies requires \(2\) cups of flour. How many cups of flour are needed to make \(30\) cookies?
Set up a proportion: \(\frac{12 \text{ cookies}}{2 \text{ cups}} = \frac{30 \text{ cookies}}{x \text{ cups}}\). Cross-multiply: \(12x = 30 \times 2\), so \(12x = 60\). Divide by \(12\) to get \(x = 5\) cups of flour.
The cost of a phone plan is in direct proportion to the amount of data used. If \(5\) GB of data costs \(\$30\), how much would \(8\) GB of data cost?
Set up a proportion: \(\frac{5 \text{ GB}}{\$30} = \frac{8 \text{ GB}}{x}\). Cross-multiply: \(5x = 30 \times 8\), so \(5x = 240\). Divide by \(5\) to get \(x = \$48\).
Solve for \(z\): \(\frac{z-2}{4} = \frac{7}{2}\)
Cross-multiply: \(2(z-2) = 4(7)\). Distribute on the left side: \(2z - 4 = 28\). Add \(4\) to both sides: \(2z = 32\). Divide by \(2\) to get \(z = 16\).
How is the graph of a direct proportion represented on a coordinate plane?
The graph of a direct proportion (\(y=kx\)) is a straight line that passes through the origin \((0,0)\). The slope of the line is equal to the constant of proportionality, \(k\).
