Proportional Relationships - ISEE Middle Level Quantitative Reasoning

1. What is $y$ if $y=kx$, $k=\frac{3}{4}$, and $x=20$?

   $y=15$. Substituting into the equation yields $y = (3/4) \times 20 = 15$.

2. What is $x$ if $ \frac{x}{5}=\frac{12}{15}$?

   $x=4$. Cross-multiplying the proportions $15x = 60$ and dividing by 15 yields $x=4$.

3. What percent is $18$ of $72$ using $ \frac{\text{part}}{\text{whole}}=\frac{p}{100}$?

   $25\%$. Setting up $18/72 = p/100$ and solving gives $p = (18/72) \times 100 = 25$.

4. What is the direct method to find a missing value in $ \frac{a}{b}=\frac{x}{d}$?

   $x=\frac{ad}{b}$ (with $b\ne^0$). Solving for the missing value uses cross-multiplication to equate products, yielding $x$ as the product of $a$ and $d$ divided by $b$.

5. What is the unit rate if $18$ items cost $24?

   $\$\frac{4}{3}$ per item. Dividing total cost by number of items gives the rate of $24/18$, which simplifies to $4/3$ per item.

6. What is the equation of a proportional relationship using constant $k$?

   $y=kx$. A proportional relationship is modeled by this linear equation where $y$ varies directly with $x$ through the constant $k$.

7. What is $x$ if $ \frac{9}{12}=\frac{x}{20}$?

   $x=15$. Cross-multiplying yields $180 = 12x$, and dividing by 12 gives $x=15$.

8. What is the missing value $x$ in the scale ratio $ \frac{3}{5}=\frac{12}{x}$?

   $x=20$. Cross-multiplying yields $3x = 60$, so dividing by 3 gives $x=20$.

9. Which option shows ratios that are proportional: A) $ \frac{2}{3}$ and $ \frac{8}{12}$, B) $ \frac{2}{3}$ and $ \frac{8}{10}$?

   A) $\frac{2}{3}$ and $\frac{8}{12}$. Simplifying $8/12$ to $2/3$ shows equivalence to the first ratio, while $8/10$ simplifies to $4/5$, which differs.

10. Identify whether $ \frac{5}{6}$ and $ \frac{20}{24}$ are proportional.

    Yes, because $5\cdot24=6\cdot20$. The cross-products are equal at 120, confirming the ratios are proportional.

11. Find $y$ if $y$ is proportional to $x$ and $y=14$ when $x=8$, then $x=20$.

    $y=35$. The constant $k=14/8=7/4$, so for $x=20$, $y=20 \times (7/4)=35$.

12. What is the percent proportion used to find a part of a whole?

    $\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$. The percent proportion sets the ratio of part to whole equal to the percent over 100 to solve for unknowns.

13. What is $k$ if $y$ is proportional to $x$ and $(x,y)=(6,15)$?

    $k=\frac{5}{2}$. Dividing $y$ by $x$ gives the constant $k=15/6$, which simplifies to $5/2$.

14. What is the direct method to find a missing value in $ \frac{a}{b}=\frac{c}{x}$?

    $x=\frac{bc}{a}$ (with $a\ne^0$). Cross-multiplication equates $a x = b c$, so isolating $x$ gives the product of $b$ and $c$ divided by $a$.

15. What is the unit rate for a ratio written as $ \frac{a}{b}$?

    $\frac{a}{b}$ per $1$ (the value when the second quantity is $1$). The unit rate simplifies the ratio to express the value of the first quantity when the second is exactly 1.

16. What is $x$ if $ \frac{3}{8}=\frac{6}{x}$?

    $x=16$. Cross-multiplying results in $3x = 48$, so dividing by 3 provides $x=16$.

17. What is the cost of $15$ items if $6$ items cost $10 (constant unit price)?

    $25. The unit price is $10/6 = 5/3$, so for 15 items the cost is $15 \times (5/3) = 25$.

18. What is the constant of proportionality $k$ in $y=kx$?

    $k=\frac{y}{x}$ for $x\ne^0$. The constant of proportionality $k$ represents the fixed ratio of $y$ to $x$ in a direct proportion, defined as their quotient when $x$ is not zero.

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

    Proportional if and only if $ad=bc$ (with $b\ne^0$ and $d\ne^0$). The cross-products test checks equality by verifying if the product of the numerator of one ratio and the denominator of the other equals the reverse, excluding zero denominators.

20. What is $x$ if $ \frac{7}{x}=\frac{21}{12}$?

    $x=4$. Cross-multiplying gives $84 = 21x$, so dividing by 21 solves for $x=4$.

21. What is the distance traveled in $5$ hours at a constant speed of $42$ miles per $3$ hours?

    $70$ miles. The speed is $42/3 = 14$ mph, so in 5 hours the distance is $5 \times 14 = 70$ miles.

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

    They are equal: $ \frac{a}{b}=\frac{c}{d}$ with $b\ne^0$ and $d\ne^0$. Two ratios are proportional when they express the same relationship, making their fractions equivalent provided the denominators are not zero.

23. What point must be on the graph of any proportional relationship $y=kx$?

    $(0,0)$. The graph of $y=kx$ is a straight line through the origin, so it always passes through $(0,0)$ regardless of $k$.

24. What is $x$ if $y=kx$, $k=\frac{5}{3}$, and $y=25$?

    $x=15$. Rearranging $y = kx$ to $x = y/k$ and substituting gives $x = 25 / (5/3) = 15$.

25. Identify the proportional equation for a constant ratio $ \frac{y}{x}=k$.

    $y=kx$. This equation captures the direct variation where $y$ is always a constant multiple $k$ of $x$.

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