Word problems that add or subtract fractions
Takoda ran \tfrac{4}{5} mile and Remi ran \tfrac{1}{2} mile. Nothing was taken away from anyone, so why is “how much farther did Takoda run?” a subtraction?
Where you are 5 in Adding and subtracting fractions
When you reach for it
Phuong, Remi and Takoda swim on the same team, and on weekends they enter a road race. Their days are full of fractions: \tfrac{3}{4} mile of warm-up, \tfrac{2}{5} mile of kicking, a water bottle \tfrac{7}{8} full, a race that is 1\tfrac{1}{2} miles long.
You already know how to add and subtract fractions with different denominators: rename both in the same size of piece, then count (Adding and subtracting fractions with unlike denominators). This lesson is about the step before that. A story does not come with a plus or minus sign. You have to read what is happening and decide which one the story needs. You reach for this lesson whenever a situation gives you fractions or mixed numbers and asks a question about them.
The lesson has three parts, one for each day:
- Part 1: joining and taking away, and the bar picture that shows which one a story is.
- Part 2: comparing two amounts, and finding a missing part. Both can sound like adding, and both are solved by subtracting.
- Part 3: stories with two steps, and an estimate to judge the answer.
The procedure
Part 1: joining and taking away
Every story in this lesson is about parts and a whole. Some amount is the whole, and it is made of two parts. The question asks for one of the three numbers, and the other two are given.
- If both parts are known and the whole is not, the parts are joined. You add.
- If the whole and one part are known, the other part is what is left. You subtract.
A bar shows this. Draw one long bar for the whole and split it into the two parts. Write the numbers you know on the bar, and a question mark on the one you want. The picture tells you what to do. When the question mark sits over the whole bar, add the parts. When it sits on one part, take the known part away from the whole.
In the bar above, Takoda jogs \tfrac{3}{4} mile and then walks \tfrac{1}{6} mile. Both parts are on the bar, and the question mark is over the whole trip, so the trip is \tfrac{3}{4} + \tfrac{1}{6} mile. Example 1 works it out.
A story that takes something away has the same bar, read the other way. Phuong’s bottle starts \tfrac{7}{8} full and she drinks some. Now the whole is known, the part she drinks is known, and the question mark sits on the part still in the bottle. That is a subtraction (Example 2).
Two more facts matter before any numbers are written down.
Both amounts need the same whole. \tfrac{3}{4} mile and \tfrac{1}{6} mile can be joined, because both are fractions of one mile. \tfrac{3}{4} of a mile and \tfrac{1}{6} of an hour cannot: they measure different things, and their sum is not an amount of anything.
The answer carries the unit. If the parts are in miles, the whole is in miles. If they are fractions of a bottle, the answer is a fraction of the bottle. Writing the unit is part of answering the question.
A single word in the story does not decide. The words “in all” can sit in a story that joins, and also in “how much more does she need in all?”, which asks for a missing part. Read the whole story, draw the bar, and let the question mark decide.
Part 2: comparing, and missing parts
Two kinds of story ask for a difference without anything being taken away.
Comparing. Takoda runs \tfrac{4}{5} mile and Remi runs \tfrac{1}{2} mile. “How much farther did Takoda run?” No one loses any distance. But draw two bars that start at the same edge, one for each runner. The question asks for the extra piece at the end of Takoda’s bar, the piece Remi’s bar would need to match it.
Think of Takoda’s bar as a whole made of two parts: Remi’s length, and the extra. The whole is known and one part is known, so the extra is the other part. That is subtraction, the same as in Part 1: \tfrac{4}{5} - \tfrac{1}{2} (Example 3). Any question about how far apart two amounts are asks for that extra piece, whatever words it uses.
Missing parts. The road race is 1\tfrac{1}{2} miles long and Phuong has run \tfrac{5}{8} mile. How far does she still have to go? The story is about adding: what she has run, plus what she will run, makes the whole race.
The question mark is a part, not the whole. To find a missing part, take the known part away from the whole. Subtraction undoes addition: if \tfrac{5}{8} and the missing part make 1\tfrac{1}{2}, then taking \tfrac{5}{8} back off 1\tfrac{1}{2} leaves exactly the missing part. A number line shows it as a gap.
So the story is written as an addition with a blank, and it is solved by a subtraction (Example 4). The addition gives a free check at the end: add the answer back to the known part and you must land on the whole.
“More” alone does not decide. Look at these two sentences.
- “Takoda swims \tfrac{1}{4} mile more than Phuong.” If you know Phuong’s distance, Takoda’s is Phuong’s plus the extra. You add.
- “Phuong swims \tfrac{1}{4} mile more than Takoda.” If you know Phuong’s distance, Takoda’s is Phuong’s without the extra. You subtract.
Both use the words “more than”. What decides is which person has the longer bar and which length the question asks for. Example 5 works both.
Part 3: two steps, and an estimate
Some stories need two operations. A relay team shares a course, and the last runner covers what the first two did not. A swimmer swims on two days and then compares her total with a friend’s. The bar picture still works. Draw it with every part the story names, mark what is known, and look at the question mark. Often one step builds a number (a total, or what is left) that the second step needs.
Order matters only when the story fixes it. Taking away two parts one after the other, 2 - \tfrac{5}{8} - \tfrac{3}{4}, leaves the same amount as taking away their sum at once, 2 - \left(\tfrac{5}{8} + \tfrac{3}{4}\right), because both remove the same two pieces from the same whole. Use whichever is easier to rename.
Before working, estimate. Replace each fraction with a nearby 0, \tfrac{1}{2} or 1 and do the story with those numbers (Estimating with fractions). An estimate does two jobs here. It tells you roughly where the exact answer should land, and it can catch a wrong choice of operation. If a story asks how far Remi swam when Phuong swam farther, the answer must be less than Phuong’s distance. If your answer comes out bigger, you added when the story needed a subtraction.
Worked examples
Examples 1 and 2 belong to Part 1, Examples 3, 4 and 5 to Part 2, and Examples 6 and 7 to Part 3.
Joining two parts of a trip
Takoda jogs \tfrac{3}{4} mile to warm up for the road race. Then he walks \tfrac{1}{6} mile to the start line. How far does he go before the race starts?
SolutionThe trip has two parts, the jog and the walk, and both are known. The question asks for the whole trip. That is the bar at the top of the lesson, with the question mark over the whole bar, so the parts are joined.
The parts are fourths and sixths. The least common denominator is 12, because 12 is the smallest number that both 4 and 6 divide into. Rename each part in twelfths.
Takoda goes \tfrac{11}{12} mile before the race. Check it with sense: \tfrac{3}{4} is less than 1, and a short walk of \tfrac{1}{6} mile does not quite reach the next whole, so a total just under 1 mile fits.
Taking away part of a bottle
Phuong’s water bottle is \tfrac{7}{8} full at the start of the swim meet. Between races she drinks \tfrac{1}{3} of a bottle. How full is the bottle now?
SolutionHere the whole is what she started with, \tfrac{7}{8} of a bottle. One part is what she drank. The question asks for the part still in the bottle. The whole and one part are known, so take the known part away.
Both amounts are fractions of the same bottle, so they can be subtracted. Eighths and thirds share no smaller common denominator than 24, so rename both in twenty-fourths.
The bottle is \tfrac{13}{24} full. Check by adding back: \tfrac{13}{24} + \tfrac{8}{24} = \tfrac{21}{24}, which is \tfrac{7}{8}, the amount she started with. The answer is a bit more than half a bottle, which fits: she started with almost a full bottle and drank about a third.
How much farther
At the road race, Takoda runs \tfrac{4}{5} mile and Remi runs \tfrac{1}{2} mile. How much farther does Takoda run than Remi?
SolutionNothing is taken away from either runner, so look at the bars instead. Both bars start at the same line, as in the second figure. Takoda’s bar is longer. Split it into two parts: a part as long as Remi’s run, and the extra. The extra is what the question asks for. Takoda’s whole bar and the part matching Remi’s are both known, so the extra is a missing part, and a missing part is found by subtracting.
The least common denominator of 5 and 2 is 10.
Takoda runs \tfrac{3}{10} mile farther. Check: Remi’s \tfrac{5}{10} mile and the extra \tfrac{3}{10} mile make \tfrac{8}{10} mile, which is Takoda’s \tfrac{4}{5} mile.
How far is still to go
The road race is 1\tfrac{1}{2} miles long. Phuong has run \tfrac{5}{8} mile so far. How much farther does she have to run to finish?
SolutionThe story joins two parts: what Phuong has run and what she still has to run make the whole race. Written as an addition, the unknown is one of the parts.
A missing part is the whole with the known part taken away, as on the number line in Part 2. So this addition story is solved by a subtraction.
Write 1\tfrac{1}{2} as a fraction. One whole is \tfrac{2}{2}, so 1\tfrac{1}{2} = \tfrac{3}{2}. Since 2 \times 4 = 8, halves can be cut into eighths: \tfrac{3}{2} = \tfrac{12}{8}.
Phuong has \tfrac{7}{8} mile still to run. Check with the addition: \tfrac{5}{8} + \tfrac{7}{8} = \tfrac{12}{8} = 1\tfrac{1}{2}, the full race. Other ways to subtract a fraction from a mixed number are in Subtracting mixed numbers.
The same words, two operations
At swim practice, Phuong swims \tfrac{2}{5} mile of freestyle.
(a) Takoda swims \tfrac{1}{4} mile more than Phuong. How far does Takoda swim?
(b) On another day, Phuong again swims \tfrac{2}{5} mile, and this time she swims \tfrac{1}{4} mile more than Remi. How far does Remi swim?
Solution(a) Takoda swims more, so his bar is the longer one. It is made of Phuong’s length and the extra \tfrac{1}{4} mile. Both parts are known and the question asks for the whole bar, so add. The least common denominator of 5 and 4 is 20.
Takoda swims \tfrac{13}{20} mile.
(b) Now Phuong swims more, so her bar is the longer one. It is made of Remi’s length and the extra \tfrac{1}{4} mile. The whole bar (\tfrac{2}{5}) and one part (the extra) are known, and the question asks for the other part, Remi’s length. So subtract.
Remi swims \tfrac{3}{20} mile.
Both stories say “\tfrac{1}{4} mile more than”. In (a) the person asked about has the longer bar, so the answer is bigger than \tfrac{2}{5}. In (b) the person asked about has the shorter bar, so the answer is smaller than \tfrac{2}{5}. Each answer passes that test.
Three runners share a course
Phuong, Remi and Takoda run a 2 mile relay. Phuong runs the first \tfrac{5}{8} mile and Remi runs the next \tfrac{3}{4} mile. Takoda runs the rest. How far does Takoda run? Estimate first.
SolutionEstimate. \tfrac{5}{8} is a little more than \tfrac{1}{2}, and \tfrac{3}{4} is halfway between \tfrac{1}{2} and 1. Together they are about 1\tfrac{1}{4} to 1\tfrac{1}{2} miles. That leaves Takoda about \tfrac{1}{2} to \tfrac{3}{4} mile.
Exact. The bar is the whole course, 2 miles, in three parts. Two parts are known and Takoda’s is the question mark, so his part is the whole minus the two known parts. Taking both away at once means first joining them. The least common denominator of 8 and 4 is 8.
Phuong and Remi run \tfrac{11}{8} miles together. The whole course is 2 = \tfrac{16}{8}.
Takoda runs \tfrac{5}{8} mile, which lands inside the estimate. Check by joining all three parts: \tfrac{5}{8} + \tfrac{6}{8} + \tfrac{5}{8} = \tfrac{16}{8} = 2 miles, the whole course.
Who swam farther over two days
Phuong swims \tfrac{7}{12} mile on Monday and \tfrac{3}{4} mile on Tuesday. Takoda swims 1\tfrac{1}{6} miles in all over the same two days. Who swims farther over the two days, and by how much?
SolutionEstimate. \tfrac{7}{12} is just over \tfrac{1}{2}, and \tfrac{3}{4} is between \tfrac{1}{2} and 1, so Phuong’s total is somewhere around 1\tfrac{1}{4} miles. Takoda’s 1\tfrac{1}{6} miles is close to that. Expect a small difference, and no way to tell the winner from the estimate alone.
Step 1: Phuong’s total. Her two days are two parts and the question needs the whole, so add. The least common denominator of 12 and 4 is 12.
Step 2: compare. Now there are two totals, 1\tfrac{1}{3} and 1\tfrac{1}{6}. Both have 1 whole, so compare the fraction parts: \tfrac{1}{3} = \tfrac{2}{6}, which is more than \tfrac{1}{6}. Phuong swims farther. “By how much” asks for the extra piece on the longer bar, which is a subtraction.
Phuong swims \tfrac{1}{6} mile farther than Takoda. The difference is small, as the estimate said. Adding mixed numbers like Phuong’s total is the subject of Adding mixed numbers.
Boundaries
- One whole for every amount. Fractions can be joined or compared only when they are fractions of the same whole, measured in the same unit. \tfrac{1}{2} mile and \tfrac{1}{2} hour do not add to anything, and \tfrac{1}{2} of a small bottle and \tfrac{1}{3} of a large one are not \tfrac{5}{6} of either bottle.
- A whole that is not given. Some stories never name the whole, because it is “one” of something: “she has learned \tfrac{3}{5} of the routine” leaves 1 - \tfrac{3}{5} to learn. The whole is 1 only when the fractions are parts of one thing.
- Larger minus smaller. Every subtraction here takes a smaller amount from a larger one. A question that needs the other order (more water drunk than there was) describes something that cannot happen with these quantities. Numbers below zero come later (Operations with rational numbers).
- Joining and separating only. Some stories with fractions are not about parts and a whole at all. “\tfrac{3}{4} of the team is at the race, and \tfrac{1}{2} of those are running” takes a fraction of a fraction, which is multiplication (A fraction times a fraction), not subtraction.
Where this goes next
The same reading of parts and wholes carries every later story with numbers in it. Stories with larger mixed numbers use the methods of Adding mixed numbers and Subtracting mixed numbers. Stories where a fraction of an amount is taken call for multiplying fractions (A fraction times a fraction).
Later, the blank in “\tfrac{5}{8} + \;?\; = 1\tfrac{1}{2}” gets a letter, and the story becomes an equation such as x + \tfrac{5}{8} = 1\tfrac{1}{2}. Solving it by taking \tfrac{5}{8} off both sides is exactly the move of Part 2 (One-step equations).
Exercises
Part 1: joining and taking away
For each story, say whether parts are being joined or one part is being taken away, then answer the question.
- Remi jogs \tfrac{1}{3} mile and then walks \tfrac{1}{2} mile to cool down. How far does she go in all?
- Phuong has \tfrac{3}{4} hour before her race. She spends \tfrac{1}{2} hour stretching. How much time is left before the race?
- At practice, Takoda swims \tfrac{1}{4} mile of kicking and \tfrac{3}{5} mile of freestyle. How far does he swim?
- Volunteers paint a chalk line along the race course. Before lunch they paint \tfrac{5}{12} of the line, and after lunch another \tfrac{1}{3} of it. How much of the line is painted? Give the answer in simplest form.
- The team cooler holds \tfrac{5}{6} gallon of water. The swimmers drink \tfrac{1}{4} gallon. How much water is left?
- Phuong runs 1\tfrac{1}{2} miles in the morning and \tfrac{2}{5} mile in the afternoon. How far does she run that day?
- Write a story about the road race that is solved by \tfrac{4}{5} - \tfrac{1}{3}, and solve it.
- A bar for a 1 mile run is split into two parts. One part is labeled \tfrac{3}{8} mile and the other is labeled with a question mark. Does the bar show joining or taking away? Find the missing part.
Part 2: comparing and missing parts
- Phuong runs \tfrac{4}{5} mile and Remi runs \tfrac{3}{4} mile. How much farther does Phuong run?
- Takoda’s warm-up takes \tfrac{5}{12} hour and Phuong’s takes \tfrac{1}{4} hour. How much longer is Takoda’s warm-up?
- The road race is 1\tfrac{1}{4} miles long. Remi has run \tfrac{2}{5} mile. How much farther does she have to run?
- Remi’s goal is to swim \tfrac{9}{10} mile today. She has swum \tfrac{2}{5} mile. How much more does she need to swim?
- Takoda runs \tfrac{3}{5} mile. Phuong runs \tfrac{1}{6} mile more than Takoda. How far does Phuong run?
- Remi swims \tfrac{1}{6} mile less than Phuong. Remi swims \tfrac{2}{3} mile. How far does Phuong swim?
- Phuong runs \tfrac{7}{8} mile. That is \tfrac{1}{6} mile more than Remi runs. How far does Remi run?
- Takoda had some water in his bottle. He poured in \tfrac{1}{2} liter more, and now he has \tfrac{7}{8} liter. How much water was in the bottle before?
Part 3: two steps, with an estimate
Estimate each answer first, then find it exactly.
- Remi’s bottle has \tfrac{3}{4} liter of water. She drinks \tfrac{1}{2} liter, then refills it with \tfrac{2}{3} liter. How much water is in the bottle now?
- A 1 mile relay is shared by three runners. Phuong runs \tfrac{1}{4} mile and Remi runs \tfrac{2}{5} mile. Takoda runs the rest. How far does Takoda run?
- Phuong wants to swim 1\tfrac{1}{4} miles this week. She swims \tfrac{1}{2} mile on Monday and \tfrac{3}{8} mile on Tuesday. How much farther does she need to swim this week?
- Remi runs \tfrac{11}{12} mile. Takoda runs \tfrac{1}{6} mile less than Remi. A friend works out that Takoda ran 1\tfrac{1}{12} miles. Use an estimate to decide whether that can be right, then find how far Takoda ran.
- In two races, Phuong runs \tfrac{1}{2} mile and \tfrac{2}{3} mile. In the same two races, Remi runs \tfrac{3}{4} mile and \tfrac{1}{3} mile. Who runs farther in all, and by how much?
Reasoning
- Write two short stories that both use the words “\tfrac{1}{3} mile more than” and a distance of \tfrac{1}{2} mile: one solved by adding, one solved by subtracting. Explain what makes them different.
- “How much farther does she have to run?” is a story about adding: what she has run plus what is left makes the race. Explain why it is solved by subtracting.
- Takoda says that \tfrac{3}{4} mile of swimming and \tfrac{1}{2} hour of running make 1\tfrac{1}{4} of exercise. Explain what is wrong.
- Phuong runs \tfrac{7}{8} mile, which is \tfrac{1}{4} mile farther than Remi runs. A friend says Remi ran 1\tfrac{1}{8} miles. Without finding the exact answer, explain how you can tell the friend chose the wrong operation. Then say which operation is right and what it gives.
Answers
- Joining. \tfrac{5}{6} mile. \tfrac{1}{3} + \tfrac{1}{2} = \tfrac{2}{6} + \tfrac{3}{6} = \tfrac{5}{6}.
- Taking away. \tfrac{1}{4} hour. \tfrac{3}{4} - \tfrac{1}{2} = \tfrac{3}{4} - \tfrac{2}{4} = \tfrac{1}{4}.
- Joining. \tfrac{17}{20} mile. \tfrac{1}{4} + \tfrac{3}{5} = \tfrac{5}{20} + \tfrac{12}{20} = \tfrac{17}{20}.
- Joining. \tfrac{3}{4} of the line. \tfrac{5}{12} + \tfrac{1}{3} = \tfrac{5}{12} + \tfrac{4}{12} = \tfrac{9}{12}, and dividing top and bottom by 3 gives \tfrac{3}{4}.
- Taking away. \tfrac{7}{12} gallon. \tfrac{5}{6} - \tfrac{1}{4} = \tfrac{10}{12} - \tfrac{3}{12} = \tfrac{7}{12}.
- Joining. 1\tfrac{9}{10} miles. The fraction parts join: \tfrac{1}{2} + \tfrac{2}{5} = \tfrac{5}{10} + \tfrac{4}{10} = \tfrac{9}{10}, and the 1 whole stays, so 1\tfrac{1}{2} + \tfrac{2}{5} = 1\tfrac{9}{10}.
- Answers vary. One story: “The race course is \tfrac{4}{5} mile long. Remi has run \tfrac{1}{3} mile. How far is left?” The answer is \tfrac{7}{15} mile: \tfrac{4}{5} - \tfrac{1}{3} = \tfrac{12}{15} - \tfrac{5}{15} = \tfrac{7}{15}. Any story where \tfrac{4}{5} is the whole (or the longer amount) and \tfrac{1}{3} is the known part works.
- Taking away. The whole (1 mile) and one part are known, so the missing part is the whole with the known part taken away. \tfrac{5}{8} mile: 1 - \tfrac{3}{8} = \tfrac{8}{8} - \tfrac{3}{8} = \tfrac{5}{8}.
- \tfrac{1}{20} mile. Comparing, so subtract: \tfrac{4}{5} - \tfrac{3}{4} = \tfrac{16}{20} - \tfrac{15}{20} = \tfrac{1}{20}.
- \tfrac{1}{6} hour. \tfrac{5}{12} - \tfrac{1}{4} = \tfrac{5}{12} - \tfrac{3}{12} = \tfrac{2}{12}, which simplifies to \tfrac{1}{6}.
- \tfrac{17}{20} mile. A missing part: \tfrac{2}{5} + \;?\; = 1\tfrac{1}{4}. Since 1\tfrac{1}{4} = \tfrac{5}{4} = \tfrac{25}{20} and \tfrac{2}{5} = \tfrac{8}{20}, the gap is \tfrac{25}{20} - \tfrac{8}{20} = \tfrac{17}{20}.
- \tfrac{1}{2} mile. A missing part: \tfrac{9}{10} - \tfrac{2}{5} = \tfrac{9}{10} - \tfrac{4}{10} = \tfrac{5}{10} = \tfrac{1}{2}.
- \tfrac{23}{30} mile. Phuong has the longer bar, made of Takoda’s length and the extra, so add: \tfrac{3}{5} + \tfrac{1}{6} = \tfrac{18}{30} + \tfrac{5}{30} = \tfrac{23}{30}.
- \tfrac{5}{6} mile. “Less” here means Phuong has the longer bar: Remi’s length plus the extra. Add: \tfrac{2}{3} + \tfrac{1}{6} = \tfrac{4}{6} + \tfrac{1}{6} = \tfrac{5}{6}.
- \tfrac{17}{24} mile. Phuong has the longer bar, and Remi’s length is that bar without the extra. Subtract: \tfrac{7}{8} - \tfrac{1}{6} = \tfrac{21}{24} - \tfrac{4}{24} = \tfrac{17}{24}.
- \tfrac{3}{8} liter. The story is ? + \tfrac{1}{2} = \tfrac{7}{8}, a missing part: \tfrac{7}{8} - \tfrac{1}{2} = \tfrac{7}{8} - \tfrac{4}{8} = \tfrac{3}{8}.
- \tfrac{11}{12} liter. Estimate: \tfrac{3}{4} - \tfrac{1}{2} is about a quarter liter, and adding about \tfrac{2}{3} gives a little under 1 liter. Exact: \tfrac{3}{4} - \tfrac{1}{2} = \tfrac{1}{4}, then \tfrac{1}{4} + \tfrac{2}{3} = \tfrac{3}{12} + \tfrac{8}{12} = \tfrac{11}{12}.
- \tfrac{7}{20} mile. Estimate: \tfrac{1}{4} and \tfrac{2}{5} together are a bit more than \tfrac{1}{2}, so Takoda runs a bit less than \tfrac{1}{2}. Exact: \tfrac{1}{4} + \tfrac{2}{5} = \tfrac{5}{20} + \tfrac{8}{20} = \tfrac{13}{20}, and 1 - \tfrac{13}{20} = \tfrac{7}{20}.
- \tfrac{3}{8} mile. Estimate: she has swum about \tfrac{1}{2} + \tfrac{1}{2} = 1 mile, so about \tfrac{1}{4} mile is left. Exact: \tfrac{1}{2} + \tfrac{3}{8} = \tfrac{4}{8} + \tfrac{3}{8} = \tfrac{7}{8}; 1\tfrac{1}{4} = \tfrac{10}{8}, and \tfrac{10}{8} - \tfrac{7}{8} = \tfrac{3}{8}.
- It cannot be right. Takoda runs less than Remi, and Remi runs less than 1 mile, so Takoda’s distance is less than 1 mile, not 1\tfrac{1}{12}. (That number is what adding gives.) Takoda runs \tfrac{3}{4} mile: \tfrac{11}{12} - \tfrac{1}{6} = \tfrac{11}{12} - \tfrac{2}{12} = \tfrac{9}{12} = \tfrac{3}{4}.
- Phuong, by \tfrac{1}{12} mile. Phuong: \tfrac{1}{2} + \tfrac{2}{3} = \tfrac{3}{6} + \tfrac{4}{6} = \tfrac{7}{6} = 1\tfrac{1}{6}. Remi: \tfrac{3}{4} + \tfrac{1}{3} = \tfrac{9}{12} + \tfrac{4}{12} = \tfrac{13}{12} = 1\tfrac{1}{12}. Compare: \tfrac{14}{12} - \tfrac{13}{12} = \tfrac{1}{12}.
- Answers vary. Adding: “Remi runs \tfrac{1}{2} mile. Takoda runs \tfrac{1}{3} mile more than Remi. How far does Takoda run?” (\tfrac{5}{6} mile). Subtracting: “Phuong runs \tfrac{1}{2} mile, which is \tfrac{1}{3} mile more than Remi runs. How far does Remi run?” (\tfrac{1}{6} mile). In the first, the person asked about has the longer bar, so the answer is the known length plus the extra. In the second, the person asked about has the shorter bar, so the answer is the known length without the extra.
- The distance left is one of the two parts, and the whole race is known. If what she has run plus the missing part makes the whole, then taking what she has run back off the whole leaves exactly the missing part. Subtraction undoes addition, so a missing part in an addition is found by subtracting.
- The two amounts are fractions of different wholes: one is part of a mile of distance, the other part of an hour of time. Fractions can be joined only when they are parts of the same whole, so there is no unit for the “1\tfrac{1}{4}” to be. The two amounts can only be reported separately.
- Remi runs less than Phuong, because Phuong runs farther. So Remi’s distance must be less than \tfrac{7}{8} mile, and 1\tfrac{1}{8} is more than \tfrac{7}{8}. The friend added. Remi’s distance is Phuong’s longer bar without the extra, so subtract: \tfrac{7}{8} - \tfrac{2}{8} = \tfrac{5}{8} mile.