Solving Word Problems Using Rational Equations Practice

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/4+1/6\) job per hour. Its reciprocal is \(\frac{12}{5}\) hours, which is less than the faster solo time \(4\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/6+1/8\) job per hour. Its reciprocal is \(\frac{24}{7}\) hours, which is less than the faster solo time \(6\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/5+1/10\) job per hour. Its reciprocal is \(\frac{10}{3}\) hours, which is less than the faster solo time \(5\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/8+1/12\) job per hour. Its reciprocal is \(\frac{24}{5}\) hours, which is less than the faster solo time \(8\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/9+1/18\) job per hour. Its reciprocal is \(6\) hours, which is less than the faster solo time \(9\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/10+1/15\) job per hour. Its reciprocal is \(6\) hours, which is less than the faster solo time \(10\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/12+1/20\) job per hour. Its reciprocal is \(\frac{15}{2}\) hours, which is less than the faster solo time \(12\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/7+1/14\) job per hour. Its reciprocal is \(\frac{14}{3}\) hours, which is less than the faster solo time \(7\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/6+1/9\) job per hour. Its reciprocal is \(\frac{18}{5}\) hours, which is less than the faster solo time \(6\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/15+1/25\) job per hour. Its reciprocal is \(\frac{75}{8}\) hours, which is less than the faster solo time \(15\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/8+1/24\) job per hour. Its reciprocal is \(6\) hours, which is less than the faster solo time \(8\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/10+1/30\) job per hour. Its reciprocal is \(\frac{15}{2}\) hours, which is less than the faster solo time \(10\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/12+1/18\) job per hour. Its reciprocal is \(\frac{36}{5}\) hours, which is less than the faster solo time \(12\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/14+1/21\) job per hour. Its reciprocal is \(\frac{42}{5}\) hours, which is less than the faster solo time \(14\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/16+1/24\) job per hour. Its reciprocal is \(\frac{48}{5}\) hours, which is less than the faster solo time \(16\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/18+1/27\) job per hour. Its reciprocal is \(\frac{54}{5}\) hours, which is less than the faster solo time \(18\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/20+1/30\) job per hour. Its reciprocal is \(12\) hours, which is less than the faster solo time \(20\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/9+1/12\) job per hour. Its reciprocal is \(\frac{36}{7}\) hours, which is less than the faster solo time \(9\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/15+1/20\) job per hour. Its reciprocal is \(\frac{60}{7}\) hours, which is less than the faster solo time \(15\).

Convert each time to a rate, add the rates, and invert for one whole job.

Their combined rate is \(1/21+1/28\) job per hour. Its reciprocal is \(12\) hours, which is less than the faster solo time \(21\).

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(2/6=\frac{1}{3}\), leaving \(\frac{2}{3}\). Dividing that remainder by the combined rate \(\frac{5}{12}\) gives \(\frac{8}{5}\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(2/8=\frac{1}{4}\), leaving \(\frac{3}{4}\). Dividing that remainder by the combined rate \(\frac{7}{24}\) gives \(\frac{18}{7}\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(3/10=\frac{3}{10}\), leaving \(\frac{7}{10}\). Dividing that remainder by the combined rate \(\frac{3}{10}\) gives \(\frac{7}{3}\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(4/12=\frac{1}{3}\), leaving \(\frac{2}{3}\). Dividing that remainder by the combined rate \(\frac{5}{24}\) gives \(\frac{16}{5}\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(3/9=\frac{1}{3}\), leaving \(\frac{2}{3}\). Dividing that remainder by the combined rate \(\frac{5}{18}\) gives \(\frac{12}{5}\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(5/15=\frac{1}{3}\), leaving \(\frac{2}{3}\). Dividing that remainder by the combined rate \(\frac{1}{6}\) gives \(4\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(4/14=\frac{2}{7}\), leaving \(\frac{5}{7}\). Dividing that remainder by the combined rate \(\frac{3}{14}\) gives \(\frac{10}{3}\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(6/16=\frac{3}{8}\), leaving \(\frac{5}{8}\). Dividing that remainder by the combined rate \(\frac{7}{48}\) gives \(\frac{30}{7}\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(5/18=\frac{5}{18}\), leaving \(\frac{13}{18}\). Dividing that remainder by the combined rate \(\frac{1}{6}\) gives \(\frac{13}{3}\) additional hours.

Separate the solo stage, subtract its completed work from 1, then divide the remainder by the new combined rate.

A first completes \(8/20=\frac{2}{5}\), leaving \(\frac{3}{5}\). Dividing that remainder by the combined rate \(\frac{7}{60}\) gives \(\frac{36}{7}\) additional hours.

Attach units to distinguish rate from time.

One tank divided by 5 hours is \(1/5\) tank per hour.

Multiply the fractional rate by elapsed time.

Work equals rate times time: \((1/8)(3)=3/8\).

Convert time units before multiplying by an hourly rate.

Ninety minutes is \(3/2\) hours, so work is \((1/6)(3/2)=1/4\).

Add rates, not completion times.

Independent simultaneous contributions add as fractions of the same job per hour.

Write a work contribution for each worker.

Each contribution is rate times time: \(x/a\) and \(x/b\); their sum is one job.

Subtract completed work from one whole job.

The whole project is 1, so \(1-2/5=3/5\).

Use the faster-together sanity check before detailed arithmetic.

Together they must be faster than the 6-hour worker, so 8 hours is impossible.

Read whether the question asks for final-stage or total time.

Total elapsed time is \(2+3/2=7/2\) hours.

Create one rate term for every active machine.

Each machine contributes work during the same interval, so all three rates add.

Match the right side to the stated number of whole jobs.

The target work is three whole jobs, so contributions sum to 3.

Use a common denominator to add every active rate.

The combined rate is \(1/6+1/8+1/12=3/8\), so time is \(1/(3/8)=8/3\) hours.

Assign a negative rate to work that undoes progress.

The drain has negative contribution: net rate \(1/4-1/12=1/6\), so the tank fills in 6 hours.

Do not assume the target work is always one job.

Rate is work divided by time: \(3/5\) book per hour.

Divide the target work by the rate.

Time is work divided by rate: \(9div(3/5)=15\) hours.

Compute the first contribution, subtract from one, then divide by the next rate.

A completes \(4/10=2/5\), leaving \(3/5\). B needs \((3/5)/(1/6)=18/5\) hours.

Subtract the known rate from the combined rate, then invert.

The unknown rate is \(1/4-1/12=1/6\), so its solo time is 6 hours.

Use remaining capacity as the target work.

The remaining amount is \(2/3\); dividing by \(1/8\) gives \((2/3)(8)=16/3\) hours.

Convert only after solving if the model used hours consistently.

Multiply hours by 60: \((5/2)(60)=150\) minutes.

Apply contextual constraints after algebraic solving.

An elapsed duration cannot be negative; revisit the model or reject a nonphysical algebraic root.

Verify work contributions in the original contextual model.

Reconstructing every stage tests the equation, units, and interpretation simultaneously.