What is a fraction word problem

Here are some examples and solutions of fraction word problems.
The first example is a one-step word problem.
The second example shows how blocks can be used to help illustrate the problem.
The third example is a two-step word problem.

More examples and solutions using the bar modeling method to solve fraction word problems are shown in
the videos. The bar modeling method, also called tape diagrams, are used in Singapore Math and the Common
Core.

Example:
Martha spent
of her allowance on food and shopping. What fraction of her allowance had she left?

Solution:

She had
of her allowance left.

Example:
of a group of children were girls. If there were 24 girls, how many children were there in the group?

Solution:

3 units = 24
1 unit = 24 ÷ 3 = 8
5 units = 5 × 8 = 40

There were 40 children in the group.

Example:
Sam had 120 teddy bears in his toy store. He sold

of them at $12 each. How much did he receive?

Solution:
Step 1: Find the number of teddy bears sold.

He sold 80 teddy bears.

Step 2: Find how much money he received.
80 × 12 = 960

He received $960.

Bar Modeling With Fractions

Examples:

1. Grace thought that a plane journey would take 7/10 hr but the actual journey took 1/5 hr longer.
   How long did the actual journey take?
2. Timothy took 2/3 hr to paint a portrait. This was 1/3 hr shorter than the time he took to paint
   scenery. How long did he take to paint scenery?

• Show Video Lesson

How To Solve Real World Problems Involving Multiplication Of Fractions By Using Visual Fraction Models Or Equations To Represent The Problem?

Examples:

1. At the animal shelter 4/6 of the animals are cats. Of the cats 1/2 are male. What fraction of the
   animals at the shelter are male cats?
2. A taco recipe called for 2/3 cup of cheese per taco. If Andrew wanted to make 3 tacos, how much
   cheese would he need?

• Show Video Lesson

How To Solve Word Problems Using Tape Diagrams And Fraction-By-Fraction Multiplication?

Examples:

1. Ethan is icing 30 cupcakes. He spreads mint icing on 1/5 of the cupcakes and chocolate on 1/2 of the
   remaining cupcakes. The rest will get vanilla frosting. How many cupcakes have vanilla frosting?
2. Maddox puts 1/4 of her lawn-mowing money in savings and uses 1/2 of the remaining money to pay back
   her sister. If she has $15 left, how much did she have at first?

• Show Video Lesson

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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Fraction word problems are an important piece in teaching fractions. Fractions are a key component of the National Curriculum throughout primary, with children introduced to the concept in Year 1 and continuously building upon their knowledge and understanding, through to Year 6 and beyond.

The abstract nature of fractions makes them one of the more difficult areas of maths for children to understand. It is essential that they are taught through the use of concrete resources and visual images, but also through a link to the real world. Word problems are a great way to help children develop this understanding and should be used alongside any fluency work the children do.

To help you with this, we have put together a collection of 20 word problems which can be used with children from Year 2 to Year 6.

Fraction word problems in the national curriculum

Fractions in KS1

Pupils are first introduced to fractions in Year 1, where they begin to recognise a half and quarter of a shape or quantity. As they progress through Key Stage 1, they are introduced to a wider range of fractions.

In Year 2 pupils need to be able to recognise, find, name and write fractions frac{1}{3}, frac{1}{4}, frac{2}{4} and frac{3}{4} of lengths, shape, a set of objects or quantity.

Fractions in lower KS2

Once pupils move into Key Stage 2, there is an increased focus on fractions, with a significant period of time dedicated to developing students’ knowledge and building on previous learning from Year 3 through to Year 6.

In Year 3, pupils are taught to count in tenths; recognise, find and write fractions of a discrete set of objects; understand the term ‘unit’ and ‘non-unit’ fractions; recognise equivalent fractions, compare and order fractions and add/subtract fractions with the same denominator. At this stage, pupils also begin to solve simple problems involving fractions.

As pupils progress into Year 4, they are also introduced to the concept of decimals alongside the work on fractions. They build upon their understanding of equivalent fractions and adding/subtracting fractions (still with the same denominator.

They also learn to count up and down in hundredths, recognise and write decimals equivalent to frac{1}{4}, frac{1}{2} and frac{3}{4} and begin to solve problems with increasingly harder fractions.

Fractions in upper KS2

As pupils continue through the school, into upper Key Stage 2, they continue to build on their learning. They are introduced to mixed numbers and improper fractions. They are also introduced to the concept of percentages alongside fractions and decimals.

In Year 5, children build on the knowledge gained in Year 4, comparing and ordering fractions; recognising equivalent fractions; addition and subtraction of fractions (with different denominators, but multiples of the same number). Fractions appear in addition and subtraction word problems.

They are also introduced to the multiplication of fractions; converting between improper fractions and mixed numbers and the link between fractions, decimals and percentages. By Year 5 pupils will be completing more complex problems, including multi-step word problems and multiplication word problems.

In Year 6, pupils consolidate and build upon the knowledge of fractions they have gained so far. Pupils continue to add and subtract fractions (with different denominators and mixed numbers). They also develop their understanding of how to multiply fractions and begin to divide fractions by whole numbers, combining fraction and division word problems.

Pupils also continue to build upon their understanding of the equivalences between fractions, decimals and percentages and solve increasingly complex one-step, two-step and multi-step percentage word problems.

Why are word problems important for children’s understanding of fractions?

Word problems, alongside the use of concrete resources and visual images, help children to make sense of the abstract nature of fractions. 

For example, if children were presented with the question frac{6}{8} ÷ 3, some might find this difficult to access. If children are given it as a word problem, alongside a visual image, it becomes much more accessible.

The above calculation as a word problem:

 ‘3 children shared it{frac{6}{8}} of a pizza, how much did each child get?’

Children can see and understand the concept of 3 children sharing 6 slices of pizza, compared to just seeing the written calculation.

How to teach fraction word problem solving in primary school

As with all word problems, the first thing children need to do is read the question carefully and make sure they understand what is being asked. They then have to decide what calculation is needed and whether they can represent it pictorially.

Here is an example:

Ben has a bag of 48 sweets.

He gives frac{3}{8} of his bag to his brother

How many sweets does he have left?

How to solve:

What do you already know?

• There are 48 sweets altogether. 
• The denominator is 8, so we know we need to divide the 48 by 8. This will tell us how many sweets in frac{1}{8} .
• The numerator is 3, therefore we need to multiply the answer by 3, to tell us how many sweets are frac{3}{8} .
• The last step of the problem is to find out how many sweets Ben has left.
• We can either calculate frac{5}{8} , as we know Ben must have frac{5}{8} left, or we subtract the number of sweets in frac{3}{8} from the total of 48.

How can this be drawn/represented pictorially?

• We can draw a bar model, split into 8 sections. 
• The 48 sweets are then shared between the 8 sections (each worth frac{1}{8} ). 
• We can see that frac{3}{8} of the 48 sweets equals 18 sweets. 
• We can then either subtract 18 from 48, to work out how many sweets were left. 48 – 18 = 30
• Alternatively, we can calculate how many sweets are in frac{5}{8} , which is 30.

Word problems are tied in throughout Third Space Learning’s online one-to-one tuition programmes. Tutors work with students to break down word problems and identify the operations needed to solve problems.

Fraction word problems for primary 

The following section lays out the types of fraction word problems that you can expect to see by year group in primary school. 

At Third Space Learning we incorporate word problems into our one-to-one online tutoring. With each lesson designed to suit the needs of each individual student, our tutoring programme aims to deepen mathematical understanding and confidence in maths. 

Fraction word problems are a great way to strengthen students’ conceptual understanding of fractions and grow their problem solving skills. 

Fraction word problems for Year 2

Word problems in Year 2 involve recognising and naming simple fractions and beginning to understand the concept of equivalence.

Fraction word problems for Year 3

Word problems for Year 3 include proper fractions, fractions of amounts, simple equivalent fractions and adding and subtracting fractions with the same denominator. 

Fraction word problems for Year 4

Word problems for year 4 include equivalent fractions, fractions of quantities (including non-unit fractions and adding/subtracting fractions with the same denominator. Money word problems often appear in Year 4.

Fraction word problems for Year 5

Word problems for Year 5 include more complex equivalent fractions, problems involving mixed numbers and improper fractions, adding and subtracting fractions (with different denominators) and multiplying fractions.

Fraction word problems for Year 6

Word problems for year 6 include adding and subtracting fractions with different denominators and mixed numbers, multiplying fractions, dividing fractions by whole numbers and finding equivalent fractions, decimals and percentages.

More word problems resources

Third Space Learning offers a wide range of practice word problems for all primary year groups and covering a large number of topics, including; time word problems, ratio word problems, addition word problems and subtraction word problems.

Word Problems on Fractions: A fraction is a mathematical expression for a portion of a whole. Each portion acquired when we divide the entire whole into parts is referred to as a fraction. When we divide a pizza into parts, for example, each slice represents a fraction of the whole pizza. Fractions are subjected to a variety of operations, including addition, subtraction, multiplication, and division. Fractions are used in many real-life situations.

This article will outline how to construct and solve fraction word problems. Students will come across fraction word problems with answers, fraction problem solving and dividing fractions word problems. It is advisable to practice all the problems thoroughly before attempting the exam. Keep reading to know more about word problems on fractions,, definition, types, solved examples and many more

Definition of Fractions

A fraction is a number that is used to expresses a part per whole. Each part obtained when we divide the whole into several parts is called the fraction.

Example: When we cut an apple into two-part, then each part represents the fraction (left(frac{1}{2}right)) of the apple.

A fraction consists mainly of two parts, one is the numerator, and the other one is the denominator. The upper part or topmost part of the fraction is called the numerator, and the bottom part or below part is called the denominator.

Example:

We have mainly three types of fractions: proper fractions, improper fractions, and mixed fractions. They are categorised by the relationship between the numerator and denominator of the fractions.

Word Problems on Fractions

The fraction problem solving consist of a few sentences describing a real-life scenario where a mathematical calculation of fraction formulas are used to solve a problem.

Some of the word problems on fractions that uses fraction formula are listed below:

1. Word problems on simplification of fractions
2. Word problems on addition and subtraction of fractions
3. Word problems on multiplication of fractions
4. Word problems on dividing fractions
5. Word problems on fractions, percentages, decimals.

Word Problems on Simplifications of Fractions

A fraction in which the numerator and the denominator have no common factor other than “one” is said to be the simplest form of fractions.

Example: Divya took (8) apples from the bucket of (24) apples. Find the fraction of apples taken by the Divya?
The fraction of apples taken by Divya (=frac{8}{24}) and its simplest form is (frac{1}{3})

Word Problems on Addition of Fractions

To add the like fractions (Fractions with the same denominators), keep the denominator the same and add the numerator values of the given fractions.

To add the unlike fractions (fractions with different denominators), convert the denominators of the given fractions equal to L.C.M of their denominators. Now add the numerator value and take the denominator of the resultant as L.C.M.

Example: Sahana bought (frac{1}{4} mathrm{~kg}) of apples and (frac{1}{2} mathrm{~kg}) of oranges from the shop. Total how many fruits she bought?
The total fruits bought by Sahana (=frac{1}{2}+frac{1}{4}=frac{1 times 2+1}{4}=frac{3}{4} mathrm{~kg})

Word Problems on Subtraction of Fractions

To subtract the like fractions (Fractions with the same denominators), keep the denominator the same and find the difference of the numerator values of the given fractions.

To subtract the unlike fractions (fractions with different denominators), convert the denominators of the given fractions equal to L.C.M of their denominators. Now find the difference of the numerator value and take the denominator of the resultant as L.C.M.

Example: Keerthi travelled (frac{2}{5} mathrm{~km}) to school. While returning home, she stopped at her friend’s house at a distance of (frac{1}{3} mathrm{~km}). Find the remaining distance?
The remaining distance needs to be travelled (=frac{2}{5}-frac{1}{3}=frac{(2 times 3)-(1 times 5)}{5 times 3}=frac{6-5}{15}=frac{1}{15} mathrm{~km})

Word Problems on Multiplication of Fractions

To multiply the two or more fractions, find the product of numerators of the given fractions and the product of the denominators of the given fractions separately.

Example: Keerthi had (Rs.10000), and she had donated (frac{1}{10}) of the money to the Oldage home. How much amount did she donate?
The amount Keerthi donated (=frac{1}{10} times Rs.10000= Rs. 1000)

Word Problems on Division of Fractions

The division of fractions is nothing but multiplying the first fraction with the reciprocal of the second fraction. The reciprocal of the fraction is a fraction obtained by interchanging the numerator and denominator.

Example: The area of the rectangle is (frac{15}{4} mathrm{~cm}^{2}), whose length is (frac{5}{2} mathrm{~cm}). Find the width of the rectangle?
We know that area of rectangle (= text {length} times text {bredath})
And, breadth (=frac{text { area }}{text { length }}=frac{15}{frac{4}{2}}=frac{15}{4} times frac{2}{5}=frac{3}{2} mathrm{~cm}).

Word Problems on Conversion of Fractions to Percentage

We know that percentages are also fractions with the denominator equals to hundred. To convert the given fraction to a percentage, multiply it with hundred and to convert any percentage value to a fraction, divide with hundred.

Example: Keerthi ate (frac{2}{5}) of the pizza. How much percentage of pizza is eaten by Keerthi?
The percentage of pizza ate by Keerthi (=frac{2}{5} times 100 %=40 %).

Word Problems on Conversion of Fractions to Decimals

Decimal numbers are the numbers (quotient) obtained by dividing the fraction’s numerator with the given fraction’s denominator. To convert the given decimal to the fractional value by writing the given number without decimals and making the denominator equal to (1) followed by the zeroes and number of zeroes equal to the number of decimal places.

Example: Keerthi got (frac{1}{10}) of the price of a T.V. as a discount. Find the discount in decimal.
The part of the discount received by a Keerthi as a discount (=frac{1}{10}=0.1)

Solved Examples – Word Problems on Fractions

Q.1. In February (2021), a school was working only three-fourths of the total number of days in the month and the remaining number of days given as holidays. How many days did the school work in the month of February?
Ans:
The year (2021) is a non-leap year. We know that a non-leap has (28) days in February month.
So, the total number of days (=28).
Given, the school was working only three-fourths of the total number of days in the month.
The number of days school working in February month (=frac{3}{4}) of (28).
(=frac{3}{4} times 28=21) days
Hence, the school working for (21) days in the month of February for the year (2021).

Q.2. Keerthi needs (1 frac{1}{2}) cups of sugar for baking a cake. She decided to make (6) cakes for her friends. How many cups of sugar did she need for making the (6) cakes?
Ans:
Given, Keerthi needs (1 frac{1}{2}) cup of sugar to make a cake.
The total cups of sugar required to make 6 cakes is calculated by multiplying the sugar needed for one cake with the number of cakes that needs to be prepared by Keerthi and is given by (1 frac{1}{2} times 6)
Convert the above-mixed fraction to an improper fraction by multiplying the denominator with the whole and add to the numerator keeping the same denominator as
(1 frac{1}{2}=frac{(text { whole×denominator })+text { numerator })}{text { denominator }}=frac{(1 times 2)+1}{2}=frac{3}{2})
The total cups of sugar needed for making (6) cakes (=frac{3}{2} times 6=9)
Hence, Keerthi needs (9) cups of sugar to make (6) cakes.

Q.3. An oil container contains (7 frac{1}{2}) litres of oil which are poured into (2 frac{1}{2}) litres bottles. How many bottles are needed to fill (7 frac{1}{2}) litres of oil?
Ans:
Given, a container holds total oil of (7 frac{1}{2}) litres, and the total amount held by each bottle is (2 frac{1}{2}) litres.
Consider the number of bottles required is (x).
From the given question, the total oil in the container is equal to the product of oil in each bottle and the number of bottles required.
(Rightarrow 7 frac{1}{2}=x times 2 frac{1}{2})
(Rightarrow frac{15}{2}=x times frac{5}{2})
(Rightarrow 15=5 x)
(Rightarrow x=frac{15}{5}=3)
Therefore, (3) bottles are required to fill the total oil in the container.

Q.4. A square garden has the area (frac{36}{25} ,text {sq.ft}). Find the side of the square garden.
Ans:
Given the area of the square garden is (frac{36}{25} ,text {sq.ft}).
Let the length of the side of the square garden is (a) fts.
We know that area of the square ( = {rm{side}} times {rm{side}} = {a^2})
Thus, (a^{2}=frac{36}{25})
(Rightarrow a=sqrt{frac{36}{25}}=frac{sqrt{36}}{sqrt{25}}=frac{6}{5}) feet.
Hence, the length of the side of the square garden is (frac{6}{5}) feet.

Q.5. At a party, total (280) ice-creams are prepared. Four-seventh of them is eaten by the children. Find the ice-creams eaten by the children.
Ans:
Total ice-creams prepared (=280)
Number of ice-creams eaten by children (=frac{4}{7}) of (280=frac{4}{7} times 280=160)
Hence, children ate (160) ice-creams.

Summary

In mathematics, a fraction is used to represent a piece of something larger. It depicts the whole’s equal pieces. The numerator and denominator are the two elements of a fraction. The numerator is the number at the top, while the denominator is the number at the bottom. The numerator specifies the number of equal parts taken, whereas the denominator specifies the total number of equal parts in the total.

In this article, we have studied the definitions of fractions,  different types of fractions. We also studied the word problems on fractions and their operations. This article gives the word problems on fractions, addition and subtraction of fractions, multiplication of fractions, division of fractions, the simplest form of fractions, conversion of fractions to percentage, decimals etc., with the help of solved examples.

FAQs on Word Problems on Fractions

Here are some of most commonly asked questions on word problems on fractions.

Q.1: How do you solve word problems with fractions?

Ans: To solve word problems with fractions, first, read and write the given data. Write the mathematical form by given data and perform the operations on fractions according to the data.

Q.2: How do you write a fraction division in word problems? 

Ans: The fraction division can be written as keeping the first fraction as it is and multiplying it with the reciprocal of the second fraction.

Q.3: How do you know when to divide or multiply fractions in a word problem?

Ans: To find the product, we need to multiply and to find any one of the quantities, we need to divide.

Q.4: What is an example of a fraction word problem?

Ans: Keerthi ate 40% of the pizza. How much is part of the pizza eaten by Keerthi.

Q.5: What is a fraction?

Ans: A fraction is a number that is used to express a part per whole.

Learn About Conversion Of Fractions

We hope this detailed article on Word Problem on Fractions helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.

Stay tuned to Embibe to learn more important concepts

A great variety of fraction word problems with solutions. These word problems will help you find out when you need to add, subtract, multiply or divide fractions.

1. Sarah knows 8 of the 25 students in her class. Write a fraction showing the number of students Sarah knows.

2. Peter correctly solved 18 out of the 20 math problems the teacher gave him. Write a fraction showing the number of math problems Peter solved correctly.

Solution

18/20

3. Darlene ate 3/5 of her pizza. How much pizza can Darlene share with her sister?

Solution

1 represents the entire pizza. To find how much pizza is left, just do 1 — 3/5

1 — 3/5 = 1/1 — 3/5 = 5/5 — 3/5 = 2/5

Darlene can share 2/5 of the pizza.

4. Last week, Steve spent 3 1/2 hours playing soccer and 4 1/2 hours playing table tennis. How much time did Steve spend playing both sports?

5. John jogged 1 3/4 miles yesterday and 2 1/2 miles today. How many miles did John spend jogging for the past two days?

1 + 2 = 3

3/4 + 1/2 = 3/4 + 2/4 = 5/4

John jogged 3 5/4 miles for the past two days.

6. Maria bought a pizza and ate half of it. Then, she gave the leftover to his brother and his brother ate half of it. How much of the pizza did his brother eat? 

Solution

The brother ate half of half the pizza.

Half of half = 1/2 of 1/2 = 1/2 × 1/2 = 1/4

7. Anthony caught some fish that weighted 6 1/2 pounds. If he gave 4 1/4 pounds of fish to some friends, how much fish does he have for himself?

Solution

Subtract 4 1/4 from 6 1/2.

6 — 4 = 2

1/2 — 1/4 = 2/4 — 1/4 = 1/4

Anthony has 2 1/4 pounds of fish for himself.

8. 1/4 of a tablespoon of baking soda is needed to make a loaf of bread. How many loaves of bread can be made with 5 tablespoons of baking soda?

Solution

Divide 5 by 1/4

5  ÷ 1/4 = 5/1 ÷ 1/4 = 5/1  × 4/1 = 20/1 = 20

20 loaves of bread can be made.

9. A recipe calls for 2 1/8 cups of butter for a casserole. How much butter is needed to make 6 1/2 casseroles?

2  1/8  × 6 1/2 = (2 × 8 + 1)/8 × (6 × 2 + 1)/2

2  1/8  × 6 1/2 = 17/8 × 13/2 = (17 × 13)/(8 × 2)  

2  1/8  × 6 1/2 = 221/16 = 13.8125

To make 6 1/2 casseroles, the recipe will need 13.8125 cups of butter. 

10. Kevin cooked 8 1/4 pounds of meat for a dinner. If each serving is 3/4 pound, how many servings did he prepare?

Solution

Just divide 8 1/4 by 3/4

8 1/4 ÷ 3/4 = (8 × 4 + 1)/4 ÷ 3/4 = 33/4 ÷ 3/4 = 33/4 × 4/3 = 132/12 = 11

Kevin prepared 11 servings.

Sometime soon I will do a series of posts on word problems, which are a common point of difficulty with students. But here is one recent example from a high school student, where language was the main difficulty, but the algebra is worth discussing as well. We’ll look a little more deeply into the problem itself, and think about its implications.

Writing and solving an equation

Here is the initial question:

1/2 of a number is 2 more than 1/3 of the number. What is 1/3 of the number?

a. 2

b. 4

c. 8

d. 12

Doctor Rick took the question, and first, as we usually do, pointed out that in order to help, we need some information about the student’s knowledge and needs. When we are just given a problem, we can’t tell what kind of help is needed; the best thing is for the student to show his work so we can see where, if at all, they made a mistake. Sometimes they turn out to be entirely correct, and it was the book that was wrong — so it would be a waste of time to write out a full explanation. Other times, the student is not learning algebra at all, so again, an algebraic solution would be wasted.

But, as we often do, he gave a suggestion to get started, in case that was all that was needed:

One way this problem might be solved is to “translate” the statement into an algebraic equation. Is that something you’ve been learning to do? For instance, give “the number” the name “x”. You’re asked to find “1/3 of the number”, which you can write as (1/3)x or x/3. How would you write “1/2 of the number”? How would you write “2 more than 1/3 of the number”?

In this case (as often happens), the student is not fluent in English, which could be the reason for not giving a full explanation. We find that in such a case, it is better to say more rather than less — the more a student says about his thinking, the more likely we will understand it, even if he doesn’t express it clearly. Here is the response (a little cleaned up):

Not good in English sorry. But I’ll try.

To be honest, I don’t know where to start because I don’t know how to approach this problem. It says “more than” so I think it’s an addition so:

x/2(2) + x/3

…

Then… I’ll need to find 1/3 of that number? I can’t understand it.

Seeing this work gave Doctor Rick a good idea of what was needed. He replied,

Thanks for the additional information about how you’re thinking. That’s the sort of thing I needed.

The problem was:

1/2 of a number is 2 more than 1/3 of the number. What is 1/3 of the number?

You said:

It says that “more than” so I think it’s an addition so:

x/2(2) + x/3

OK, I see some good things here. You translated “1/2 of a number” as x/2, and “1/3 of the number” as x/3. You’re correct that “more than” in this sentence refers to addition (sometimes, as you’ll see later, it can mean other things.)

There are two main difficulties with what you wrote. One is that it isn’t an equation — you didn’t get an equal sign (=) into it. That’s how I’d translate the word “is”. The other problem is that you are not taking the grammar of the sentence into account.

Let’s work step by step, translating one part at a time rather than trying to write the equation all at once. Here’s what I mean:

1/2 of a number is 2 more than 1/3 of the number.
_____________/ /   _______/ _______________/

      x/2       =  2     +            x/3

Wow, I’m done — I’ve written an equation! It won’t always work this neatly — in this case, the sentence is in nearly mathematical language right from the start. The same thing wouldn’t work for other problems you have sent!

OK, now you have an equation to solve. Can you find the value of x that makes the equation true? And then can you figure out what “1/3 of the number” is?

Doctor Rick has done two things here: one is to look at the whole sentence and translate it into an equation, rather than a mere expression; and the other is to not look at the whole sentence at once, but to break it into parts (according to its grammar) in order to build up the equation part by part. In this case, it was easy to do both things at once; more complicated sentences often require working on each piece separately to avoid confusion, and changing the order of parts. For example, if it had said “2 less than …”, we couldn’t just write “2 – …”, because the order would be wrong. We’d have to say x/3 – 2 instead. So matching up phrase after phrase in order is not always appropriate, but it was convenient here.

This appeared to be the hard part for this student. We like to leave as much as possible for them to solve themselves, so that they get practice doing the work themselves; here it would have been hard to explain the ideas without doing the whole translation, but we can at least leave the solving to the student, who can probably manage that. And Doctor Rick was right about that:

OMG!!! Yah, the ‘is’ is =. Here’s my solution:

x/2 = 2 + x/3

x/2 – x/3 = 2

(3x – 2x)/6 = 2

x = 12

Since the question is what is the 1/3 of the number

x/3 = 12

3(x/3 = 12)

x = 4

Aaaaaaahhhh!!! And guess what! I got it right!!!!!

Well, there are some things to clean up here, but encouragement was appropriate, and Doctor Rick responded encouragingly,

You got it correct, and you are justified in feeling good about yourself! Keep it up!

A closer look

But let’s go back and look at a few details in the solution.

First, the student used a good method, and show work quite well — not too much detail, but including the important steps. His approach was to subtract x/3 from both sides, then combine the left side into a single fraction, which reduces to x/6; then, multiplying both sides by 6, the result is x = 12.

Another approach, often taught in books and saving some work (or at least some writing, or some opportunities to make mistakes in using fractions, is to “clear fractions” before solving. The LCD of all the fractions in x/2 = 2 + x/3 is 6, so we multiply each term by 6 in order to cancel with each denominator:

x/2 = 2 + x/3

6*x/2 = 6*2 + 6*x/3

3x = 12 + 2x

Now, with no fractions to trip over, we can just subtract x from each side and solve:

3x – 2x = 12

x = 12

Second, the student was wise enough not to stop there, but to see that the question was not to give the number, but rather 1/3 of the number. So the answer is 1/3 of 12, which is 4.

But the work he wrote was wrong! It is not uncommon for a student to think correctly, but not yet be skilled in writing what he is thinking. Here, he changed x = 12 to x/3 = 12, replacing x with the x/3 that is desired; what he should have done is to divide both sides by 3, x/3 = 12/3 = 4. I can only guess that, as he approached the finish line, he started thinking and writing too fast, and stumbled. We’ll give him credit.

Let’s take one more step, and check the answer. Recall again that the problem was, “1/2 of a number is 2 more than 1/3 of the number. What is 1/3 of the number?” We found that the number is 12. What is 1/2 of the number? 6. What is 1/3 of the number? 4. Is 6 2 more than 4? Yes. If we had misinterpreted “2 more than” in translating the equation, we probably would interpret it correctly in this more familiar process, so we would have a chance to discover our error.

And, again, part of checking is seeing whether we answered the question that was asked. What is 1/3 of 12? 4. We got it.

What if he didn’t know algebra?

Let’s take this in yet another direction. I mentioned at the start that this problem might have been given to a student who is not learning algebra, though that tends to be our first assumption. How might we solve it then?

The problem, once again, was, “1/2 of a number is 2 more than 1/3 of the number. What is 1/3 of the number?” The question might lead me to focus my attention not on the number itself, but on 1/3 of the number; I may never need to know what the number itself is! Here is a bar representing the number, with 1/3 of it shaded:

+-----------+-----------+-----------+
|XXXXXXXXXXX|           |           |
+-----------+-----------+-----------+

Here I’ve split each third in halves, so I can mark 1/2 of the number:

+-----+-----+-----+-----+-----+-----+
|XXXXX|XXXXX|     |     |     |     |
+-----+-----+-----+-----+-----+-----+
_________________/

Since the 1/2 is 2 more than 1/3, that difference must be 2:

+-----+-----+-----+-----+-----+-----+
|XXXXX|XXXXX|  2  |     |     |     |
+-----+-----+-----+-----+-----+-----+
_________________/

So each piece is 2, and the shaded region (two pieces) is 4. That’s the answer.

This sort of visual thinking is taught commonly today, and is a precursor to algebra. It requires more creativity than algebra, which is just a way to turn any problem into the same kind of symbolic problem, so that you don’t need any special thinking for each problem, but can follow a nearly mindless routine. The thinking I did here involved focusing deeply on the fractions (using a common denominator, for example, without actually saying so) and on relationships. That makes it a very useful experience. Don’t rush to learn (or teach) algebra — being easier (though students are surprised to realize that), it takes away the chance to learn to think deeply!