Rates and Averages - ISEE Upper Level Quantitative Reasoning

1. A pipe fills a tank in $5$ hours and a drain empties it in $10$ hours. How long to fill with both open?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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