Proportional Relationships - ISEE Upper Level: Quantitative Reasoning
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What does it mean for two ratios to be proportional?
What does it mean for two ratios to be proportional?
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They are equal: $
\frac{a}{b} = \frac{c}{d}$ with $b \neq 0$ and $d \neq 0$. Two ratios are proportional if they represent equivalent fractions, ensuring equality holds with non-zero denominators to avoid undefined expressions.
They are equal: $
\frac{a}{b} = \frac{c}{d}$ with $b \neq 0$ and $d \neq 0$. Two ratios are proportional if they represent equivalent fractions, ensuring equality holds with non-zero denominators to avoid undefined expressions.
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Find $k$ if $y$ varies directly with $x$ and $(x,y) = (6,15)$.
Find $k$ if $y$ varies directly with $x$ and $(x,y) = (6,15)$.
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$k = \frac{5}{2}$. For direct variation, $k$ is found by dividing $y$ by $x$ from the given point.
$k = \frac{5}{2}$. For direct variation, $k$ is found by dividing $y$ by $x$ from the given point.
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Find $y$ if $y$ varies directly with $x$, $k = 3$, and $x = 8$.
Find $y$ if $y$ varies directly with $x$, $k = 3$, and $x = 8$.
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$y = 24$. Substitute $k$ and $x$ into $y = kx$ to compute $y$ in direct variation.
$y = 24$. Substitute $k$ and $x$ into $y = kx$ to compute $y$ in direct variation.
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What is the value of $x$ if $
\frac{x}{4} = \frac{9}{6}$?
What is the value of $x$ if $
\frac{x}{4} = \frac{9}{6}$?
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$x = 6$. Simplifying $\frac{9}{6}$ to $\frac{3}{2}$ and multiplying by 4 yields $x$ in the proportion.
$x = 6$. Simplifying $\frac{9}{6}$ to $\frac{3}{2}$ and multiplying by 4 yields $x$ in the proportion.
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What is the value of $x$ if $
\frac{8}{x} = \frac{12}{15}$?
What is the value of $x$ if $
\frac{8}{x} = \frac{12}{15}$?
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$x = 10$. Cross-multiplying $8 \times 15 = 12 \times x$ or inverting the simplified ratio solves for $x$.
$x = 10$. Cross-multiplying $8 \times 15 = 12 \times x$ or inverting the simplified ratio solves for $x$.
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What is the value of $x$ if $
\frac{x}{9} = \frac{14}{21}$?
What is the value of $x$ if $
\frac{x}{9} = \frac{14}{21}$?
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$x = 6$. Simplifying $\frac{14}{21}$ to $\frac{2}{3}$ and multiplying by 9 maintains the proportional relationship.
$x = 6$. Simplifying $\frac{14}{21}$ to $\frac{2}{3}$ and multiplying by 9 maintains the proportional relationship.
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What is the value of $x$ if $
\frac{3}{5} = \frac{x}{20}$?
What is the value of $x$ if $
\frac{3}{5} = \frac{x}{20}$?
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$x = 12$. Cross-multiplying the proportions $3 \times 20 = 5 \times x$ solves for $x$ by ensuring ratio equality.
$x = 12$. Cross-multiplying the proportions $3 \times 20 = 5 \times x$ solves for $x$ by ensuring ratio equality.
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Identify whether the relationship $y = 7x + 2$ is proportional.
Identify whether the relationship $y = 7x + 2$ is proportional.
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Not proportional. The added constant term in $y=7x+2$ prevents it from passing through the origin, violating direct proportionality.
Not proportional. The added constant term in $y=7x+2$ prevents it from passing through the origin, violating direct proportionality.
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Identify whether the relationship $y = 7x$ is proportional, and state $k$.
Identify whether the relationship $y = 7x$ is proportional, and state $k$.
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Proportional; $k = 7$. The equation $y=7x$ shows direct proportionality where $y$ is a constant multiple of $x$ with no additional terms.
Proportional; $k = 7$. The equation $y=7x$ shows direct proportionality where $y$ is a constant multiple of $x$ with no additional terms.
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What is the unit rate for the ratio $\frac{a}{b}$ with $b \neq 0$?
What is the unit rate for the ratio $\frac{a}{b}$ with $b \neq 0$?
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$\frac{a}{b}$ units per $1$. The unit rate expresses the quantity of the numerator per single unit of the denominator, assuming a non-zero denominator.
$\frac{a}{b}$ units per $1$. The unit rate expresses the quantity of the numerator per single unit of the denominator, assuming a non-zero denominator.
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What is the equation for an inverse variation between $y$ and $x$?
What is the equation for an inverse variation between $y$ and $x$?
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$y = \frac{k}{x}$ (equivalently $xy = k$). Inverse variation means $y$ decreases as $x$ increases, maintaining a constant product $k$.
$y = \frac{k}{x}$ (equivalently $xy = k$). Inverse variation means $y$ decreases as $x$ increases, maintaining a constant product $k$.
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Find $x$ if $y$ varies directly with $x$, $k = 4$, and $y = 52$.
Find $x$ if $y$ varies directly with $x$, $k = 4$, and $y = 52$.
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$x = 13$. Solve for $x$ by dividing $y$ by $k$ in the direct variation equation $y = kx$.
$x = 13$. Solve for $x$ by dividing $y$ by $k$ in the direct variation equation $y = kx$.
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Find $k$ if $y$ varies inversely with $x$ and $(x,y) = (5,12)$.
Find $k$ if $y$ varies inversely with $x$ and $(x,y) = (5,12)$.
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$k = 60$. For inverse variation, $k$ is the product of $x$ and $y$ from the given point.
$k = 60$. For inverse variation, $k$ is the product of $x$ and $y$ from the given point.
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Find $y$ if $y$ varies inversely with $x$, $k = 48$, and $x = 6$.
Find $y$ if $y$ varies inversely with $x$, $k = 48$, and $x = 6$.
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$y = 8$. Divide $k$ by $x$ to find $y$ using the inverse variation formula $y = \frac{k}{x}$.
$y = 8$. Divide $k$ by $x$ to find $y$ using the inverse variation formula $y = \frac{k}{x}$.
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Find $x$ if $y$ varies inversely with $x$, $k = 72$, and $y = 9$.
Find $x$ if $y$ varies inversely with $x$, $k = 72$, and $y = 9$.
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$x = 8$. Solve for $x$ by dividing $k$ by $y$ in the inverse variation equation $x = \frac{k}{y}$.
$x = 8$. Solve for $x$ by dividing $k$ by $y$ in the inverse variation equation $x = \frac{k}{y}$.
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Find the unit rate (per $1$) for $18$ miles in $3$ hours.
Find the unit rate (per $1$) for $18$ miles in $3$ hours.
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$6$ miles per hour. Divide total miles by hours to find the rate per single hour.
$6$ miles per hour. Divide total miles by hours to find the rate per single hour.
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Find the cost of $7$ items if $4$ items cost $\$10$ (constant unit price).
Find the cost of $7$ items if $4$ items cost $\$10$ (constant unit price).
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$\$17.50$. Determine unit price by dividing cost by items, then multiply by 7 for the total.
$\$17.50$. Determine unit price by dividing cost by items, then multiply by 7 for the total.
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Find the missing length if a map scale is $1$ inch : $50$ miles and the map shows $3.2$ inches.
Find the missing length if a map scale is $1$ inch : $50$ miles and the map shows $3.2$ inches.
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$160$ miles. Multiply map distance by scale factor to convert to actual miles.
$160$ miles. Multiply map distance by scale factor to convert to actual miles.
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Find the percent equivalent of the ratio $
\frac{3}{8}$.
Find the percent equivalent of the ratio $
\frac{3}{8}$.
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$37.5%$. Convert fraction to decimal by dividing 3 by 8, then multiply by 100 for percentage.
$37.5%$. Convert fraction to decimal by dividing 3 by 8, then multiply by 100 for percentage.
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Find the new amount after a $15%$ increase of $80$.
Find the new amount after a $15%$ increase of $80$.
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$92$. Calculate 15% of 80 and add to original amount for the increased value.
$92$. Calculate 15% of 80 and add to original amount for the increased value.
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Find the missing term $b$ if $a:b = 5:12$ and $a = 20$.
Find the missing term $b$ if $a:b = 5:12$ and $a = 20$.
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$b = 48$. Set up proportion $\frac{20}{b} = \frac{5}{12}$ and solve for $b$ by cross-multiplying.
$b = 48$. Set up proportion $\frac{20}{b} = \frac{5}{12}$ and solve for $b$ by cross-multiplying.
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Identify the value of $x$ that makes the ratios proportional: $9:12 = x:20$.
Identify the value of $x$ that makes the ratios proportional: $9:12 = x:20$.
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$x = 15$. Simplify 9:12 to 3:4, then multiply 20 by $\frac{3}{4}$ to find $x$.
$x = 15$. Simplify 9:12 to 3:4, then multiply 20 by $\frac{3}{4}$ to find $x$.
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What is the cross-products condition for $
\frac{a}{b} = \frac{c}{d}$?
What is the cross-products condition for $
\frac{a}{b} = \frac{c}{d}$?
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$ad = bc$ (with $b \neq 0$ and $d \neq 0$). Cross-multiplying yields equal products for proportional ratios, confirming equivalence provided denominators are non-zero.
$ad = bc$ (with $b \neq 0$ and $d \neq 0$). Cross-multiplying yields equal products for proportional ratios, confirming equivalence provided denominators are non-zero.
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What is the constant of proportionality $k$ if $y$ is proportional to $x$?
What is the constant of proportionality $k$ if $y$ is proportional to $x$?
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$k = \frac{y}{x}$ (for $x \neq 0$). The constant $k$ represents the ratio of $y$ to $x$ in direct proportionality, defined when $x$ is not zero to prevent division by zero.
$k = \frac{y}{x}$ (for $x \neq 0$). The constant $k$ represents the ratio of $y$ to $x$ in direct proportionality, defined when $x$ is not zero to prevent division by zero.
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What is the standard equation for a direct variation between $y$ and $x$?
What is the standard equation for a direct variation between $y$ and $x$?
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$y = kx$. Direct variation implies $y$ changes linearly with $x$ through a constant multiplier $k$.
$y = kx$. Direct variation implies $y$ changes linearly with $x$ through a constant multiplier $k$.
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