Work and Rate Problems - SSAT Middle Level Quantitative

1. What is the combined flow rate if one pipe fills at $f_1$ and another fills at $f_2$ (same units)?

   $f_{\text{total}} = f_1 + f_2$. Combined flow rates add when pipes fill simultaneously, increasing overall filling efficiency.

2. Pipe A fills in $6$ h and pipe B fills in $3$ h. How long to fill together?

   $2$ h. Combined rate of $1/6 + 1/3 = 1/2$ tank/h gives time as reciprocal.

3. A worker's rate increases by $25\%$. For the same job, the new time is what fraction of the old time?

   $\frac{4}{5}$. New rate $1.25$ times old inversely affects time to $1/1.25 = 4/5$.

4. Two workers together finish in $5$ h. One alone takes $10$ h. How long does the other take alone?

   $10$ h. Other's rate is combined $1/5$ minus $1/10$, yielding time as reciprocal.

5. Three identical workers each take $12$ h alone. How long do all $3$ take together for $1$ job?

   $4$ h. Three rates of $1/12$ each add to $1/4$ job/h, so time is reciprocal.

6. A worker's time decreases from $12$ h to $9$ h for the same job. What is the percent increase in rate?

   $\frac{1}{3}=33\frac{1}{3}\%$. Rate increase from $1/12$ to $1/9$ is $(1/9 - 1/12)/(1/12) = 1/3$.

7. A tank fills at $5$ gal/min and drains at $2$ gal/min. How long to net fill $30$ gal?

   $10$ min. Net rate of $3$ gal/min divides into $30$ gal for total time.

8. What is the work rate (jobs per hour) of a worker who finishes $1$ job in $t$ hours?

   $r = \frac{1}{t}$. Rate is the reciprocal of time for one job, assuming constant work efficiency.

9. What is the combined time to finish $1$ job if two workers have times $t_1$ and $t_2$ alone?

   $t = \frac{1}{\frac{1}{t_1}+\frac{1}{t_2}}$. Combined time uses the harmonic mean of individual times, derived from adding reciprocal rates.

10. Find the combined rate (jobs per hour) if A finishes in $4$ h and B finishes in $12$ h.

    $\frac{1}{3}$ job/h. Rates add as $1/4 + 1/12 = 1/3$ job/h for joint work on one job.

11. A car travels at $50$ mi/h for $\frac{3}{2}$ h. What distance does it travel?

    $75$ mi. Distance applies speed times time, with fractional hours handled proportionally.

12. State the formula for rate $r$ when work $W$ is completed in time $t$.

    $r = \frac{W}{t}$. Rate is calculated by dividing the total work by the time, reflecting efficiency per unit time.

13. A worker completes $\frac{3}{5}$ of a job in $6$ h. What is the time for the full job?

    $10$ h. Rate from $3/5$ in $6$ h is $1/10$ job/h, so full job time is reciprocal.

14. Identify the proportional relationship between time $t$ and rate $r$ for a fixed amount of work.

    $t \propto \frac{1}{r}$. Time and rate are inversely proportional for fixed work, as doubling rate halves time.

15. A machine runs at $15$ units/min for $8$ min. How many units are produced?

    $120$ units. Production uses rate times time, applying direct proportionality for constant output.

16. What is the combined work rate if one worker has rate $r_1$ and another has rate $r_2$?

    $r_{\text{total}} = r_1 + r_2$. Combined rates add when workers contribute simultaneously to the same job without interference.

17. Find the time to complete $1$ job together if A takes $6$ h alone and B takes $3$ h alone.

    $2$ h. Combined rate is $1/6 + 1/3 = 1/2$ job/h, so time is reciprocal for one job.

18. A drain empties a full tank in $4$ h. What fraction of the tank is emptied in $30$ min?

    $\frac{1}{8}$. Rate of $1/4$ tank/h times $0.5$ h gives emptied fraction proportionally.

19. What is the distance formula in a travel rate problem using speed $v$ and time $t$?

    $d = vt$. Distance equals constant speed multiplied by time, applying rate principles to motion.

20. What fraction of a job is completed in $2$ h by a worker who needs $8$ h for $1$ job?

    $\frac{1}{4}$. Rate of $1/8$ job/h times $2$ h yields the fraction completed proportionally.

21. What is the time to finish $1$ job at a constant rate of $r$ jobs per hour?

    $t = \frac{1}{r}$. Time is the reciprocal of rate for one job, based on proportional work completion.

22. State the formula for time $t$ needed to complete work $W$ at constant rate $r$.

    $t = \frac{W}{r}$. Time is derived by dividing the total work by the constant rate, ensuring proportional reasoning.

23. What is the net flow rate if a tank fills at $f$ and drains at $d$ (same units per minute)?

    $r_{\text{net}} = f - d$. Net rate subtracts drain from fill, accounting for opposing flows in the tank.

24. State the formula for total work $W$ in terms of rate $r$ and time $t$.

    $W = rt$. The work formula multiplies the constant rate by the time taken, similar to distance equaling speed times time.

25. A pipe fills a tank in $10$ h. What fraction of the tank is filled in $3$ h?

    $\frac{3}{10}$. Fraction filled uses time divided by total time, assuming constant rate.

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