ISEE Upper Level Quantitative Reasoning Flashcards: Proportional Relationships

What this deck covers

This deck focuses on Proportional Relationships, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Upper Level Quantitative Reasoning.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: Proportional; $k = 7$. The equation $y=7x$ shows direct proportionality where $y$ is a constant multiple of $x$ with no additional terms.