Addition and subtraction of fractions

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From users who are members of Transition to university :

Chris Graham	said	Ready to use	7 years, 8 months ago
Lauren Richards	said	Needs to be tested	7 years, 8 months ago
Christian Lawson-Perfect	said	Has some problems	7 years, 8 months ago
Vicky Hall	said	Has some problems	7 years, 9 months ago

History

○

Elliott Fletcher 7 years, 8 months ago

Published this.

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Chris Graham 7 years, 8 months ago

Gave some feedback: Ready to use

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Chris Graham commented 7 years, 8 months ago

I've made some very minor changes to the advice. Looks good now.

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Lauren Richards 7 years, 8 months ago

Gave some feedback: Needs to be tested

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Christian Lawson-Perfect 7 years, 8 months ago

Gave some feedback: Has some problems

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Christian Lawson-Perfect 7 years, 8 months ago

Saved a checkpoint:

I've removed the "without a calculator" bit - I don't care how students add fractions, as long as they can do it!

The advice for parts b and c could be a lot shorter - say what's different from the previous part, how to deal with it, and then briefly show the working out.

Restore this checkpoint

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Lauren Richards 7 years, 8 months ago

Gave some feedback: Needs to be tested

○

Lauren Richards 7 years, 8 months ago

Gave some feedback: Has some problems

○

Stanislav Duris commented 7 years, 8 months ago

I haven't noticed any major problems with this question, but I thought about a couple of ideas for slight adjustments.

- If you want the student to do this question without a calculator, don't you want to perhaps tell them not to use it in the question statement? You mention not using a calculator in advice but not in the question itself. Also, you tell them to simplify their answer twice in the question statement. I don't know if you've done this on purpose so they are more likely to notice it, or if this was a mistake.

- Maybe you can slightly adjust a name of this question since there is some multiplication in part c). But it does mainly focus on addition/subtraction so if it stays like it is it should be fine.

- It's a good idea to include the original question at the start of each part in the advice, but maybe it would be even nicer to use the \[ \] format to display it? And maybe also all lines in advice consisting entirely out of equations could be changed into that format. This would make it look a bit neater, since this topic requires a lot of steps and it may be easy to get lost in.

- Although this isn't the question with a main focus on GCD and LCM and I am not sure how you would write this in advice, I still think more advice (maybe just 1 or 2 general sentences with an example different from a question that wouldn't be with variables) on how to find these would help students with simplifying fractions since I think that is the big part of the question. However, if you have included this in some of your other questions about fractions, this isn't essential.

- Also, in Advice at the end of each part you wrote "Using this value to simplify, the final answer is ...", maybe specifying you're dividing both numerator and denominator with this value would help someone struggling with simplification and can't see how it works clearly.

- You keep using the plural form of denominator when saying "keeping the denominators the same" - I don't know if this is better than just using a singular as this is when the two denominators merge into one.

- In the numerator part, the third line consists of just a fullstop from the previous line. I know this will probably differ depending on the device the student is using, but if this is the standard layout it would be great if you could try to fix this somehow. It's a silly thing but it may be distracting. If it's displayed differently on each computer, I'm sorry and I'll shut up.

- When the greatest common divisor is 1 and there is no simplification needed, the last line "Simplifying using...." doesn't really make sense, so it could be changed to be conditionally visible only if gcd ≠ 1, if you're not sure how to do that I can help. However, it can often be very laggy to work with this and Numbas sometimes crashes when you do this, so it's up to you.

Ultimately I think this question does the job well. Good job designing/defining all the variables cause it looks tricky.

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Lauren Richards commented 7 years, 9 months ago

I have enacted all the changes you suggested and particularly ensured that all the fractions in the question and advice will cancel down to their simplest form by making them all coprime.

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Lauren Richards 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Vicky Hall 7 years, 9 months ago

Gave some feedback: Has some problems

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Vicky Hall commented 7 years, 9 months ago

This question has largely the same issues as the other fraction questions I have already commented on. Have a look through and check that:

1) You're using \displaystyle where necessary.

2) There are no unsimplified fractions in the questions.

3) There are no unnecessary colons in the advice.

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Lauren Richards 7 years, 9 months ago

Gave some feedback: Needs to be tested

○

Lauren Richards 7 years, 9 months ago

Created this.

Name	Status	Author	Last Modified
Addition and subtraction of fractions	Ready to use	Lauren Richards	20/11/2019 14:32
Jo-Ann's copy of Addition and subtraction of fractions	Ready to use	Jo-Ann Lyons	14/01/2019 15:43
surbhi's copy of Addition and subtraction of fractions	draft	surbhi gautam	14/01/2019 15:47
Addition und Subtraktion von Brüchen	Ready to use	Ulrich Görtz	26/11/2020 10:32
NG3 Add Fractions	Ready to use	Oliver Spenceley	10/04/2024 09:35
NG4 Subtract fractions	Ready to use	Andrew Neate	10/04/2024 09:35
AE4 - Multiplication of algebraic fractions 1	Ready to use	Andrew Neate	28/02/2025 13:53

There are 39 other versions that do you not have access to.

Question statement

Evaluate the following additions and subtractions, giving each fraction in its simplest form.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	3
a_coprime	number	1
b	integer	6
b_coprime	number	2
gcd_ab	number	3
c	integer	2
c_coprime	number	2
d	integer	15
d_coprime	number	15
gcd_cd	number	1
lcm	number	30
a_coprimed_coprime	number	15
c_coprimeb_coprime	number	4
lcm_b	number	15
lcm_d	number	2
alcm_b	number	15
clcm_d	number	4
alcmclcm	number	19
gcd	number	1
num	number	19
denom	number	30

			Name	Type	Generated Value

f	integer	7
f_coprime	number	7
g	integer	2
g_coprime	number	2
gcd_fg	number	1
h	integer	9
h_coprime	number	9
j	integer	10
j_coprime	number	10
gcd_hj	number	1
lcm2	number	10
lcm2_g	number	5
flcm2_g	number	35
lcm2_j	number	1
hlcm2_j	number	9
flcmhlcm	number	26
twolcm2	number	20
num2unsim	number	46
gcd2	number	2

			Name	Type	Generated Value

k	number	0.89
gcd_k100	number	1
k_simp	number	89
simp	number	100
l	integer	2
l_coprime	number	1
m	integer	10
m_coprime	number	5
gcd_lm	number	2
n	integer	3
n_coprime	number	1
o	integer	12
o_coprime	number	4
gcd_no	number	3
gcd1	number	100
gcd3	number	100
term1	number	1
term2	number	20
term3	number	25
num1	number	84
gcd_numgcd3	number	4

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1..5)

JME function reference

Description

PART A variable a - random number between 1 and 5.

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

a_coprime
gcd_ab

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Part b) Gap-fill
1. Gap Gap 0. Number entry
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Part c) Gap-fill
1. Gap Gap 0. Number entry
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$\displaystyle\frac{\var{a_coprime}}{\var{b_coprime}}+\frac{\var{c_coprime}}{\var{d_coprime}}=$ Gap 0. Gap 1.

Advice

a)

$\displaystyle\frac{\var{a_coprime}}{\var{b_coprime}}+\frac{\var{c_coprime}}{\var{d_coprime}}$

To add or subtract fractions, we need to have a common denominator on both fractions.

To get a common denominator, we need to find the lowest common multiple of the two denominators.

The lowest common multiple of $\var{b_coprime}$ and $\var{d_coprime}$ is $\var{lcm}.$

This will be the new denominator, and we need to multiply each fraction individually to ensure we get this denominator.

For $\displaystyle\frac{\var{a_coprime}}{\var{b_coprime}}$ , we need to multiply the fraction by $\displaystyle\frac{\var{lcm_b}}{\var{lcm_b}}$ to give $\displaystyle\frac{\var{alcm_b}}{\var{lcm}}.$

For $\displaystyle\frac{\var{c_coprime}}{\var{d_coprime}}$ , we need to multiply the fraction by $\displaystyle\frac{\var{lcm_d}}{\var{lcm_d}}$ to give $\displaystyle\frac{\var{clcm_d}}{\var{lcm}}.$

Now that we have each fraction in terms of a common denominator, we can now add the fractions together.

$\displaystyle\frac{\var{alcm_b}}{\var{lcm}}+\frac{\var{clcm_d}}{\var{lcm}}=\frac{(\var{alcm_b}+\var{clcm_d})}{\var{lcm}}=\frac{\var{alcmclcm}}{\var{lcm}}.$

From this, we can try to simplify the result down by finding the greatest common divisor of the numerator and denominator and dividing the whole fraction by this amount.

The greatest common divisor of $\var{alcmclcm}$ and $\var{lcm}$ is $\var{gcd}.$

Simplifying using this value gives a final answer of $\displaystyle\frac{\var{num}}{\var{denom}}.$

Therefore, the expression cannot be simplified further, and $\displaystyle\frac{\var{num}}{\var{denom}}$ is the final answer.

b)

$\displaystyle\frac{\var{f_coprime}}{\var{g_coprime}}-\frac{\var{h_coprime}}{\var{j_coprime}}+2.$

The two fractions can be individually multiplied to achieve a common denominator of the lowest common multiple, $\var{lcm2}.$

$\displaystyle\frac{\var{f_coprime}}{\var{g_coprime}}$ becomes $\displaystyle\frac{\var{flcm2_g}}{\var{lcm2}}$ and $\displaystyle\frac{\var{h_coprime}}{\var{j_coprime}}$ becomes $\displaystyle\frac{\var{hlcm2_j}}{\var{lcm2}}.$

We can now subtract the second fraction from the first.

$\displaystyle\frac{\var{flcm2_g}}{\var{lcm2}}-\frac{\var{hlcm2_j}}{\var{lcm2}}=\frac{\var{flcmhlcm}}{\var{lcm2}}.$

From this, the question asks us to add $2$ . We need to change the mixed number, $2$ , into an improper fraction.

$\displaystyle2=2\bigg(\frac{\var{lcm2}}{\var{lcm2}}\bigg)=\frac{\var{twolcm2}}{\var{lcm2}}.$

We can now continue with the question.

$\displaystyle\frac{\var{flcmhlcm}}{\var{lcm2}}+\frac{\var{twolcm2}}{\var{lcm2}}=\frac{\var{num2unsim}}{\var{lcm2}}.$

We can look to simplify by dividing by the greatest common divisor of $\var{num2unsim}$ and $\var{lcm2}$ which is $\var{gcd2}.$

Simplifying by this value gives the final answer $\displaystyle\simplify{{num2unsim}/{lcm2}}.$

Therefore, no further simplification is possible, and $\displaystyle\simplify{{num2unsim}/{lcm2}}$ is the final answer.

c)

$\displaystyle\var{k}+\frac{\var{l_coprime}}{\var{m_coprime}}-\frac{\var{n_coprime}}{\var{o_coprime}}.$

We need to convert the decimal into a fraction and to do this, we need to multiply it by $10$ for every decimal place.

$\displaystyle\frac{\var{k}}{1}\times\frac{100}{100}=\frac{\var{100k}}{100}.$

We should look to simplify by dividing by the greatest common divisor which is $\var{gcd_k100}.$

Therefore, it is not possible to simplify any further, and the fraction stays as

Simplifying by this value gives the fraction

$\simplify{{{100k}}/{100}}\text{.}$

The original expression is now $\displaystyle\frac{\var{k_simp}}{\var{simp}}+\frac{\var{l_coprime}}{\var{m_coprime}}-\frac{\var{n_coprime}}{\var{o_coprime}}.$

We can multiply each fraction individually to achieve the common denominator $\var{gcd3}$ .

$\frac{\var{k_simp}}{\var{simp}}\text{ becomes }\frac{\var{k_simp*term1}}{\var{gcd3}}\text{, }\frac{\var{l_coprime}}{\var{m_coprime}}\text{ becomes }\frac{\var{l_coprime*term2}}{\var{gcd3}}\text{ and }\frac{\var{n_coprime}}{\var{o_coprime}}\text{ becomes }\frac{\var{n_coprime*term3}}{\var{gcd3}}\text{.}$

We can now complete the addition.

$\frac{\var{k_simp*term1}}{\var{gcd3}}+\frac{\var{l_coprime*term2}}{\var{gcd3}}-\frac{\var{n_coprime*term3}}{\var{gcd3}}=\frac{\var{(k_simp*term1)+(l_coprime*term2)-(n_coprime*term3)}}{\var{gcd3}}\text{.}$

We should look to simplify this fraction by dividing by the highest common divisor, $\var{gcd_numgcd3}.$

Simplifying by this value gives the final answer

Therefore, it is not possible to simplify the fraction any further and the final answer is

$\simplify{{num1}/{gcd3}}\text{.}$

Give a worked solution to the whole question.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

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Checkpoint description

Elliott Fletcher 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Lauren Richards 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Lauren Richards 7 years, 8 months ago

Lauren Richards 7 years, 8 months ago

Stanislav Duris commented 7 years, 8 months ago

Lauren Richards commented 7 years, 9 months ago

Lauren Richards 7 years, 9 months ago

Vicky Hall 7 years, 9 months ago

Vicky Hall commented 7 years, 9 months ago

Lauren Richards 7 years, 9 months ago

Lauren Richards 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

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Error

Addition and subtraction of fractions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Elliott Fletcher 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Lauren Richards 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Lauren Richards 7 years, 8 months ago

Lauren Richards 7 years, 8 months ago

Stanislav Duris commented 7 years, 8 months ago

Lauren Richards commented 7 years, 9 months ago

Lauren Richards 7 years, 9 months ago

Vicky Hall 7 years, 9 months ago

Vicky Hall commented 7 years, 9 months ago

Lauren Richards 7 years, 9 months ago

Lauren Richards 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`