ISEE Upper Level Quantitative Reasoning Flashcards: Rates And Averages

What this deck covers

This deck focuses on Rates And Averages, 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: A pipe fills a tank in $5$ hours and a drain empties it in $10$ hours. How long to fill with both open?

Answer: $10$ hours. Net rate is filling minus draining, then time is reciprocal of net rate.

Flashcard 2: What number must be added to $3,5,10$ to make the average $7$?

Answer: $10$. The required total sum for average $7$ with four numbers is $28$, so subtract the sum of three from $28$.

Flashcard 3: Find the missing value if the average of $5$ numbers is $12$ and four numbers sum to $41$.

Answer: $19$. The total sum for five numbers is $60$, so subtract the sum of four from $60$.

Flashcard 4: State the formula for average speed when traveling total distance $D$ in total time $T$.

Answer: $\text{avg speed}=\frac{D}{T}$. Average speed is defined as the total distance traveled divided by the total time taken.

Flashcard 5: What is the combined work rate if two workers have rates $r_1$ and $r_2$?

Answer: $r_\text{total}=r_1+r_2$. Combined work rates add up when workers operate simultaneously without interference.

Flashcard 6: Identify the general relationship between rate $r$, time $t$, and amount $A$.

Answer: $A=rt$. The amount produced is the product of the constant rate and the time elapsed.

Flashcard 7: What is the combined average of groups with sizes $n_1,n_2$ and averages $a_1,a_2$?

Answer: $\frac{n_1 a_1+n_2 a_2}{n_1+n_2}$. The combined average weights each group's average by its size and divides by the total size.

Flashcard 8: Find the combined average of $6$ students with average $80$ and $4$ students with average $90$.

Answer: $84$. Weight each group's sum by size and divide the total sum by the total number of students.

Flashcard 9: If a worker completes a job in $6$ hours, what fraction of the job is done in $1$ hour?

Answer: $\frac{1}{6}$. The work rate is the reciprocal of the time to complete the job.

Flashcard 10: A trip is $60$ miles at $30$ mph and then $60$ miles at $60$ mph. What is the average speed?

Answer: $40$. Calculate total distance and total time, then divide distance by time for average speed.

Flashcard 11: A rate is $45$ miles per hour. How many miles are traveled in $40$ minutes?

Answer: $30$. Multiply rate by time, converting minutes to hours.

Flashcard 12: A runner goes $3$ miles in $24$ minutes. What is the average speed in mph?

Answer: $7.5$. Convert minutes to hours and divide distance by time.

Flashcard 13: State the formula for total cost if unit price is $p$ dollars per item and quantity is $q$.

Answer: $C=pq$. Total cost is the product of the price per unit and the number of units purchased.

Flashcard 14: A store sells $3$ notebooks for $6. What is the unit rate in dollars per notebook?

Answer: $2$. Unit rate is total cost divided by number of items.

Flashcard 15: Convert a pace of $8$ minutes per mile to speed in mph.

Answer: $7.5$. Speed is $60$ minutes per hour divided by minutes per mile.

Flashcard 16: You buy $4$ lb at $3/lb and $6$ lb at $5/lb. What is the average price per pound?

Answer: $\frac{21}{5}$. Total cost divided by total weight gives the weighted average price per pound.

Flashcard 17: Worker A finishes a job in $4$ hours and Worker B in $6$ hours. How long working together?

Answer: $2.4$ hours. Add their rates and take the reciprocal to find combined time.

Flashcard 18: Machine A makes $10$ parts per hour and Machine B makes $15$ parts per hour. How many parts in $2$ hours together?

Answer: $50$. Add individual rates and multiply by time to find total output.

Flashcard 19: Find the average of $7,11,12,20$.

Answer: $12.5$. The arithmetic mean is the sum of the numbers divided by their count.

Flashcard 20: Find the new average if the average of $10$ numbers is $8$ and one number $6$ is replaced by $16$.

Answer: $9$. Replacement increases the total sum by $10$, raising the average by $1$ for $10$ numbers.

Flashcard 21: A car travels $120$ miles in $2$ hours. What is its average speed in mph?

Answer: $60$. Average speed is total distance divided by total time.

Flashcard 22: What is the work-rate formula if a job is completed in time $t$ at constant rate?

Answer: $\text{rate}=\frac{1}{t}\text{ job per unit time}$. The constant work rate is the reciprocal of the time required to complete one job.

Flashcard 23: State the formula for weighted average of values $x_i$ with weights $w_i$.

Answer: $\text{weighted avg}=\frac{\sum w_i x_i}{\sum w_i}$. The weighted average is calculated by summing the products of values and their weights, then dividing by the total weight.

Flashcard 24: A trip is $2$ hours at $30$ mph and then $1$ hour at $60$ mph. What is the average speed?

Answer: $40$. Total distance is the sum of segments, and total time is given, so average speed is distance over time.