Ratios, Rates & Proportions - ACT Math
Card 1 of 57
A recipe for $4$ servings requires $2$ cups of rice. How much rice is needed for $10$ servings?
A recipe for $4$ servings requires $2$ cups of rice. How much rice is needed for $10$ servings?
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$5$ cups (Set up proportion: $\dfrac{2}{4} = \dfrac{x}{10} \Rightarrow x = 5$)
$5$ cups (Set up proportion: $\dfrac{2}{4} = \dfrac{x}{10} \Rightarrow x = 5$)
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What is a proportion?
What is a proportion?
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An equation stating that two ratios are equal: $\dfrac{a}{b} = \dfrac{c}{d}$. Cross-product test: $\dfrac{a}{b} = \dfrac{c}{d} \iff ad = bc$
An equation stating that two ratios are equal: $\dfrac{a}{b} = \dfrac{c}{d}$. Cross-product test: $\dfrac{a}{b} = \dfrac{c}{d} \iff ad = bc$
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Convert $5$ yards to feet.
Convert $5$ yards to feet.
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$15$ feet ($5 \times 3 = 15$)
$15$ feet ($5 \times 3 = 15$)
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If $y$ varies inversely with $x$, and $y = 6$ when $x = 4$, write the equation.
If $y$ varies inversely with $x$, and $y = 6$ when $x = 4$, write the equation.
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$y = \dfrac{24}{x}$ (find $k$: $k = 6 \times 4 = 24$)
$y = \dfrac{24}{x}$ (find $k$: $k = 6 \times 4 = 24$)
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A class has $12$ boys and $18$ girls. What is the ratio of boys to girls?
A class has $12$ boys and $18$ girls. What is the ratio of boys to girls?
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$2:3$ or $\dfrac{2}{3}$ (simplify $12:18$)
$2:3$ or $\dfrac{2}{3}$ (simplify $12:18$)
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What is the difference between part-to-part and part-to-whole ratios?
What is the difference between part-to-part and part-to-whole ratios?
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Part-to-part compares two parts (e.g., boys to girls); part-to-whole compares one part to the total (e.g., boys to all students)
Part-to-part compares two parts (e.g., boys to girls); part-to-whole compares one part to the total (e.g., boys to all students)
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A $\$80$ item is on sale for $25%$ off. What is the sale price?
A $\$80$ item is on sale for $25%$ off. What is the sale price?
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$\$60$ ($80 - 0.25(80) = 80 - 20 = 60$, or $80 \times 0.75 = 60$)
$\$60$ ($80 - 0.25(80) = 80 - 20 = 60$, or $80 \times 0.75 = 60$)
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How long does it take to travel $300$ miles at $60$ mph?
How long does it take to travel $300$ miles at $60$ mph?
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$5$ hours ($t = \dfrac{d}{r} = \dfrac{300}{60} = 5$)
$5$ hours ($t = \dfrac{d}{r} = \dfrac{300}{60} = 5$)
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What is a unit rate?
What is a unit rate?
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A rate with a denominator of $1$ (e.g., $60$ miles per $1$ hour)
A rate with a denominator of $1$ (e.g., $60$ miles per $1$ hour)
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Two numbers are in the ratio $5:3$. If their sum is $64$, find the larger number.
Two numbers are in the ratio $5:3$. If their sum is $64$, find the larger number.
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$40$ (Let numbers be $5x$ and $3x$: $5x + 3x = 64 \Rightarrow 8x = 64 \Rightarrow x = 8$; larger $= 5(8) = 40$)
$40$ (Let numbers be $5x$ and $3x$: $5x + 3x = 64 \Rightarrow 8x = 64 \Rightarrow x = 8$; larger $= 5(8) = 40$)
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What is inverse variation and how do you identify the constant?
What is inverse variation and how do you identify the constant?
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$y = \dfrac{k}{x}$; to find $k$, use $k = xy$ from a known point. As $x$ increases, $y$ decreases.
$y = \dfrac{k}{x}$; to find $k$, use $k = xy$ from a known point. As $x$ increases, $y$ decreases.
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A printer prints $240$ pages in $8$ minutes. How many pages per minute?
A printer prints $240$ pages in $8$ minutes. How many pages per minute?
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$30$ pages per minute ($\dfrac{240}{8} = 30$)
$30$ pages per minute ($\dfrac{240}{8} = 30$)
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Solve for $x$: $\dfrac{3}{5} = \dfrac{x}{20}$
Solve for $x$: $\dfrac{3}{5} = \dfrac{x}{20}$
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$x = 12$ (Cross-multiply: $3 \times 20 = 5x \Rightarrow 60 = 5x \Rightarrow x = 12$)
$x = 12$ (Cross-multiply: $3 \times 20 = 5x \Rightarrow 60 = 5x \Rightarrow x = 12$)
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A recipe calls for $2$ cups of flour and $3$ cups of sugar. What is the ratio of flour to sugar?
A recipe calls for $2$ cups of flour and $3$ cups of sugar. What is the ratio of flour to sugar?
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$2:3$ or $\dfrac{2}{3}$
$2:3$ or $\dfrac{2}{3}$
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What is $30%$ of $80$?
What is $30%$ of $80$?
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$24$ ($0.30 \times 80 = 24$)
$24$ ($0.30 \times 80 = 24$)
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$15$ is what percent of $60$?
$15$ is what percent of $60$?
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$25%$ ($\dfrac{15}{60} = 0.25 = 25%$)
$25%$ ($\dfrac{15}{60} = 0.25 = 25%$)
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What is the distance formula using rate and time?
What is the distance formula using rate and time?
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$\text{distance} = \text{rate} \times \text{time}$, or $d = rt$
$\text{distance} = \text{rate} \times \text{time}$, or $d = rt$
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Convert $3.5$ hours to minutes.
Convert $3.5$ hours to minutes.
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$210$ minutes ($3.5 \times 60 = 210$)
$210$ minutes ($3.5 \times 60 = 210$)
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A store marks up items by $50%$. If the original price is $\$40$, what is the selling price?
A store marks up items by $50%$. If the original price is $\$40$, what is the selling price?
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$\$60$ ($40 + 0.50(40) = 40 + 20 = 60$, or $40 \times 1.50 = 60$)
$\$60$ ($40 + 0.50(40) = 40 + 20 = 60$, or $40 \times 1.50 = 60$)
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What is a ratio?
What is a ratio?
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A comparison of two quantities, written as $a:b$ or $\dfrac{a}{b}$
A comparison of two quantities, written as $a:b$ or $\dfrac{a}{b}$
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