Proportional Relationships - ISEE Upper Level Quantitative Reasoning

1. Identify the value of $x$ that makes the ratios proportional: $9:12 = x:20$.

   $x = 15$. Simplify 9:12 to 3:4, then multiply 20 by $\frac{3}{4}$ to find $x$.

2. What is the value of $x$ if $ \frac{3}{5} = \frac{x}{20}$?

   $x = 12$. Cross-multiplying the proportions $3 \times 20 = 5 \times x$ solves for $x$ by ensuring ratio equality.

3. What is the standard equation for a direct variation between $y$ and $x$?

   $y = kx$. Direct variation implies $y$ changes linearly with $x$ through a constant multiplier $k$.

4. Find the new amount after a $15\%$ increase of $80$.

   $92$. Calculate 15% of 80 and add to original amount for the increased value.

5. What is the value of $x$ if $ \frac{x}{9} = \frac{14}{21}$?

   $x = 6$. Simplifying $\frac{14}{21}$ to $\frac{2}{3}$ and multiplying by 9 maintains the proportional relationship.

6. Identify whether the relationship $y = 7x + 2$ is proportional.

   Not proportional. The added constant term in $y=7x+2$ prevents it from passing through the origin, violating direct proportionality.

7. Find $x$ if $y$ varies inversely with $x$, $k = 72$, and $y = 9$.

   $x = 8$. Solve for $x$ by dividing $k$ by $y$ in the inverse variation equation $x = \frac{k}{y}$.

8. What is the value of $x$ if $ \frac{x}{4} = \frac{9}{6}$?

   $x = 6$. Simplifying $\frac{9}{6}$ to $\frac{3}{2}$ and multiplying by 4 yields $x$ in the proportion.

9. Find the cost of $7$ items if $4$ items cost $10 (constant unit price).

   $17.50. Determine unit price by dividing cost by items, then multiply by 7 for the total.

10. What is the unit rate for the ratio $\frac{a}{b}$ with $b \neq 0$?

    $\frac{a}{b}$ units per $1$. The unit rate expresses the quantity of the numerator per single unit of the denominator, assuming a non-zero denominator.

11. Find the missing term $b$ if $a:b = 5:12$ and $a = 20$.

    $b = 48$. Set up proportion $\frac{20}{b} = \frac{5}{12}$ and solve for $b$ by cross-multiplying.

12. Find the missing length if a map scale is $1$ inch : $50$ miles and the map shows $3.2$ inches.

    $160$ miles. Multiply map distance by scale factor to convert to actual miles.

13. Find $y$ if $y$ varies directly with $x$, $k = 3$, and $x = 8$.

    $y = 24$. Substitute $k$ and $x$ into $y = kx$ to compute $y$ in direct variation.

14. Find $k$ if $y$ varies directly with $x$ and $(x,y) = (6,15)$.

    $k = \frac{5}{2}$. For direct variation, $k$ is found by dividing $y$ by $x$ from the given point.

15. Find the unit rate (per $1$) for $18$ miles in $3$ hours.

    $6$ miles per hour. Divide total miles by hours to find the rate per single hour.

16. What is the value of $x$ if $ \frac{8}{x} = \frac{12}{15}$?

    $x = 10$. Cross-multiplying $8 \times 15 = 12 \times x$ or inverting the simplified ratio solves for $x$.

17. What is the equation for an inverse variation between $y$ and $x$?

    $y = \frac{k}{x}$ (equivalently $xy = k$). Inverse variation means $y$ decreases as $x$ increases, maintaining a constant product $k$.

18. What does it mean for two ratios to be proportional?

    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.

19. What is the constant of proportionality $k$ if $y$ is proportional to $x$?

    $k = \frac{y}{x}$ (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.

20. Find $x$ if $y$ varies directly with $x$, $k = 4$, and $y = 52$.

    $x = 13$. Solve for $x$ by dividing $y$ by $k$ in the direct variation equation $y = kx$.

21. What is the cross-products condition for $ \frac{a}{b} = \frac{c}{d}$?

    $ad = bc$ (with $b \neq 0$ and $d \neq 0$). Cross-multiplying yields equal products for proportional ratios, confirming equivalence provided denominators are non-zero.

22. Find the percent equivalent of the ratio $ \frac{3}{8}$.

    $37.5\%$. Convert fraction to decimal by dividing 3 by 8, then multiply by 100 for percentage.

23. Find $k$ if $y$ varies inversely with $x$ and $(x,y) = (5,12)$.

    $k = 60$. For inverse variation, $k$ is the product of $x$ and $y$ from the given point.

24. Find $y$ if $y$ varies inversely with $x$, $k = 48$, and $x = 6$.

    $y = 8$. Divide $k$ by $x$ to find $y$ using the inverse variation formula $y = \frac{k}{x}$.

25. Identify whether the relationship $y = 7x$ is proportional, and state $k$.

    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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