Which expression is equivalent to 5r+11r−2?
Answers
Answer:
2(8r-1)
Step-by-step explanation:
collect like terms
5r+11r=16r
16r-2
then factor out a 2
2(8r-1)
Related Questions
three times a number subtracted by 3 is 20
Answers
Answer:
3x - 3 = 20
Step-by-step explanation:
Times is a word that represents multiplication. A number can be represented by x.
3x - 3 = 20
Solving this:
3x - 3 = 20 1. Add 3 on both sides
3x = 23 2. Divide by 3 on both sides
x = 7 2/3
friend me right quick pls
Answers
Answer:
ok ok ok i gotchuu
Step-by-step explanation:
Answer:
ok will do!
Step-by-step explanation:
You are converting the numbers shown below into scientific notation. Which of the options will be a number times 10−6
10
−
6
? Select all that apply.
A) 0.00578
A
)
0.00578
B) 0.00000445
B
)
0.00000445
C) 0.000002243
C
)
0.000002243
D) 0.000091
Answers
Answer:
a c and d
Step-by-step explanation:
help
it said line on the last one plz and its due today
Answers
Answer: D) Intersecting, but not perpendicular lines
===============================================================
Explanation:
Let's solve the second equation for y
2x - 12y = 24
-12y = 24-2x ................... subtract 2x from both sides
-12y = -2x+24
y = (-2x+24)/(-12) ............ divide both sides by -12
y = (-2x)/(-12) + 24/(-12)
y = (1/6)x - 2
The slope here is 1/6 since the last equation is in y = mx+b form.
The slope of y = 6x-2 is 6
Multiplying the two slopes leads to (1/6)*(6) = 1. Since the result is not -1, this means the lines are not perpendicular.
The lines aren't parallel either because we would need to have the two slopes be equal. Parallel lines have equal slopes but different y intercepts.
Therefore, we consider these lines to be intersecting, but not perpendicular. The single point they intersect at is the solution to the system.
6x+5 = ru what is ru?
Answers
Answer: 65
Step-by-step explanation:
Answer:
wut
Step-by-step explanation:
a) x² + 12x + 35
b) x²16x + 64
c) x²-x-12
d) x² + 3x - 54
Answers
Answer:
Step-by-step explanation:
x=5
F(x)=(x-4)^2
evaluate -2
Answers
Answer:
does that mean x=-2? because if so then:
F(x)=36
Step-by-step explanation:
(-2-4)^2
-6^2
36
F(x)=36
Let W = Span {v1, v2, .... vp}. If x is orthogonal to each of v1, v2, ...vp, show that x is orthogonal to every vector in W
Answers
The x is orthogonal to every vector w in W. This demonstrates that if x is orthogonal to each vector in the spanning set of W, it is orthogonal to every vector in W itself.
Let W = Span{v1, v2, ..., vp} be a subspace spanned by vectors v1, v2, ..., vp. We want to show that if a vector x is orthogonal to each of v1, v2, ..., vp, then x is orthogonal to every vector in W.
To prove this, consider an arbitrary vector w in W. Since W is the span of v1, v2, ..., vp, we can express w as a linear combination of these vectors, i.e., w = a1v1 + a2v2 + ... + apvp, where a1, a2, ..., ap are scalars.
Now, let's take the dot product of x with w:
x · w = x · (a1v1 + a2v2 + ... + apvp).
By the distributive property of the dot product, we have:
x · w = a1(x · v1) + a2(x · v2) + ... + ap(x · vp).
Since x is orthogonal to each of v1, v2, ..., vp, their dot products with x are all zero. Therefore, each term in the above expression becomes zero, and we have:
x · w = 0 + 0 + ... + 0 = 0.
Therefore, x is orthogonal to every vector w in W. This demonstrates that if x is orthogonal to each vector in the spanning set of W, it is orthogonal to every vector in W itself.
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Which is one of the transformations applied to the graph of f(x) = x2 to change it into the graph of g(x) = –3x2 – 36x – 60?
The graph of f(x) = x2 is made narrower.
The graph of f(x) = x2 is shifted right 6 units.
The graph of f(x) = x2 is shifted down 48 units.
The graph of f(x) = x2 is reflected over the y-axis.
Answers
Answer:
I think it is D.
Step-by-step explanation:
Answer: A
Step-by-step explanation:
A lighthouse keeper is in the top of a lighthouse 95 feet above sea level. She notes that the angle of depression to a rock jutting just above the water is 6° . How far is the rock from the lighthouse?
Answers
Answer:
808.72 ft
Step-by-step explanation:
We can create a right triangle with the following points:
- the top of the lighthouse
- the base of the lighthouse
- the rock jutting out of the water
I will label the distance from the base of the lighthouse to the rock d.
See the attached image for a detailed model.
Then, solve for the two acute angles of the triangle.
The larger acute angle (∠1) is complementary to 6°, so we can solve by setting their sum to 90°:
6° + m∠1 = 90°
m∠1 = 84°
Because this is a right triangle, the smaller acute angle (∠2) is complementary to the larger acute angle, so we can solve once again by setting their sum to 90°:
m∠1 + m∠2 = 90°
84° + m∠2 = 90°
m∠2 = 6°
We can use either of these angles in combination with the trigonometric ratio tangent to solve for the distance d. I will use tangent of ∠1.
\(\tan(84\textdegree) = \dfrac{d}{85 \, \textrm{ft}}\)
\(d = 85\tan(84\textdegree)\)
\(d \approx 808.72 \, \textrm{ft}\)
What are the coordinates of the image of the point (1-, 2) under a dilation of 3 with the origin
Answers
Answer:
The image of the point (1, -2) under a dilation of 3 is (3, -6).
Step-by-step explanation:
Correct statement is:
What are the coordinates of the image of the point (1, -2) under a dilation of 3 with the origin.
From Linear Algebra we get that dilation of a point with respect to another point is represented by:
\(\vec P' = \vec R + r\cdot (\vec P-\vec R)\) (Eq. 1)
Where:
\(\vec R\) - Reference point with respect to origin, dimensionless.
\(\vec P\) - Original point with respect to origin, dimensionless.
\(r\) - Dilation factor, dimensionless.
If we know that \(\vec R = (0,0)\), \(\vec P = (1, -2)\) and \(r = 3\), then the coordinates of the image of the original point is:
\(\vec P' = (0,0) +3\cdot [(1,-2)-(0,0)]\)
\(\vec P' = (0,0) + 3\cdot (1,-2)\)
\(\vec P' = (3,-6)\)
The image of the point (1, -2) under a dilation of 3 is (3, -6).
help i’ll mark brainliest
Answers
Answer:
12
..... ....... ......
Why do we prefer Logistic Regression over Linear Regression for classification problems? O (1), (2) & (3) The predicted values are much more interpretable as they lie in the range of 0 to 1. The sigmoid curve fits the data better than a straight line. Linear regression minimizes the mean squared error whereas logistic regression uses maximum likelihood estimation to estimate parameters.
Answers
Logistic Regression is preferred over Linear Regression for classification problems for several reasons.
Firstly, the predicted values in Logistic Regression are much more interpretable as they lie in the range of 0 to 1, which represents the probability of a certain outcome occurring. In contrast, Linear Regression predicts continuous values which are difficult to interpret as probabilities.
Secondly, the sigmoid curve used in Logistic Regression fits the data better than a straight line used in Linear Regression. This is because the sigmoid curve captures the non-linear relationships between the input variables and the output variable, which is often the case in classification problems.
Lastly, Logistic Regression uses maximum likelihood estimation to estimate parameters, whereas Linear Regression minimizes the mean squared error. Maximum likelihood estimation is a more appropriate method for estimating parameters in classification problems as it takes into account the binary nature of the output variable. Linear Regression, on the other hand, does not consider this binary nature and may not produce accurate results in classification problems.
In summary, Logistic Regression is preferred over Linear Regression for classification problems because of its interpretable predicted values, better fit to non-linear relationships, and appropriate method of estimating parameters using maximum likelihood estimation.
We prefer Logistic Regression over Linear Regression for classification problems for the following reasons:
1) Predicted values in Logistic Regression are more interpretable as they lie in the range of 0 to 1, representing probabilities of class membership, while Linear Regression predicts continuous values which may not directly indicate class membership.
2) The sigmoid curve in Logistic Regression fits the data better for classification problems, as it represents a natural threshold between classes, while a straight line from Linear Regression may not provide clear separation between classes.
3) Logistic Regression uses maximum likelihood estimation to estimate parameters, which makes it more suitable for classification problems as it finds the best fit for the probability of class membership. Linear Regression minimizes the mean squared error, which focuses on minimizing the distance between predicted and actual values, but does not directly optimize for class membership probabilities.
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If the measure of angle 4 is 43 degrees, then the measure of angle 6 is -------
degrees and the measure of angle 5 is -------- degrees. The measure of angle 8 is -------- degrees and the measure of angle 1 is --------- degrees.
Answers
Answer:
degrees and the measure of angle 5 is -------- degrees. The measure of angl
Step-by-step explanation:
Asher makes $164.00 in 8 hours at his job. If Asher works three days a week at 12 hours each day, how much will Asher make?
Answers
Answer:
738 per week
Step-by-step explanation:
20.5 per hour
246 per day
738 per week
2x+1=2(x-2) solve for x
Answers
Answer:
There is no solution to this equation
Step-by-step explanation:
2x+1=2(x-2)=
2x+1=2x-4
0x = -5
Answer is no solution
this statement is false
2x+1=2(x-2)
2x+1=2x-4
cancel equal terms
1=-4
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs 12 ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of 5 inches. Which of the following is closest to the weight, in ounces, of the second piece?
(A) 14
(B) 16
(C) 20
(D) 33.3
(E) 55.6
Answers
Answer:
D 33.3
Step-by-step explanation:
Im smart
Find all of the prime numbers between 10 and 20
Answers
Answer:
11, 13, 17, and 19!
Step-by-step explanation:
Answer:
11, 13,17 and 19
Step-by-step explanation:
prome numbers are natural numbers that are greater than 1 and not a product of two natural numbers eg 2×5 =10 so it's not a prime number 2×6 or 3×4 =12 it's not also a prime number e.t.c
the area of a rectangle shown is 44m2
3 2/3 m
x?
work out the missing length x
Answers
Step 2.1 m(t)=4cos(2π*1800Hz*t)
c(t)=5cos(2π*10.5kHz*t)
clear;
clc;
clf;
Ac=5;
Am=4;
fc=10500;
fm=1800;
t=0:0.00001:0.003;
m=Am*cos(2*pi*fm*t);
c=Ac*cos(2*pi*fc*t);
mi = Am/Ac;
s=Ac*(1+mi*cos(2*pi*fm*t)).*cos(2*pi*fc*t);
subplot(2,2,1);
plot(t,s);
xlabel('time');
ylabel('amplitude');
title('AM modulation');
subplot(2,2,4);
plot(t,m);
xlabel('time');
ylabel('amplitude');
title('Message');
subplot(2,2,2);
plot (t,c);
xlabel('time');
ylabel('amplitude');
title('Carrier');
subplot(2,2,3);
yyaxis left;
plot(t,m);
ylim([-40 40])
yyaxis right;
m(t) = Amcos(2πfmt), m=Am*cos(2*pi*fm*t),
c(t) = Ac cos(2πfct), c=Ac*cos(2*pi*fc*t),
plot(t,s);
ylim([-40 40])
title('combined message and signal');
Step 2.2 Plot the following equations by changing the variables in the step 2.1 script :
m(t) = 3cos(2π*700Hz*t)
c(t) = 5cos(2π*11kHz*t)
Having made the changes, select the correct statement regarding your observation.
a. The signal, s(t), faithfully represents the original message wave m(t)
b. The receiver will be unable to demodulate the modulated carrier wave shown in the upper left plot
c. The AM modulated carrier shows significant signal distortion
d. a and b
Step 2.3 Plot the following equations: m(t) = 40cos(2π*300Hz*t) c(t) = 6cos(2π*11kHz*t)
Select the correct statement that describes what you see in the plots:
The signal, s(t), is distorted because the AM Index value is too high
The modulated signal accurately represents m(t)
Distortion is experienced because the message and carrier frequencies are too far apart from one another
The phase of the signal has shifted to the right because AM techniques impact phase and amplitude.
Answers
Step 2.1 code is given in the question. In step 2.2, we have to change the variables, m(t) and c(t) and plot the following equations:m(t) = 3cos(2π*700Hz*t)c(t) = 5cos(2π*11kHz*t)The modified code will be:Amplitude of message signal, Am = 3 Amplitude of carrier signal, Ac = 5 Frequency of message signal, fm = 700Hz Frequency of carrier signal, fc = 11kHz.
The amplitude modulation index is given as, mi = Am/Ac = 3/5 = 0.6The modulated signal is given as,s=Ac*(1+mi*cos(2*pi*fm*t)).*cos(2*pi*fc*t);The plot of the signals can be seen below:From the plots, we can see that the signal, s(t), faithfully represents the original message wave m(t). Hence, the correct option is (a) The signal, s(t), faithfully represents the original message wave m(t).
Step 2.3 requires us to plot the following equations:m(t) = 40cos(2π*300Hz*t)c(t) = 6cos(2π*11kHz*t)The modified code will be:Amplitude of message signal, Am = 40 Amplitude of carrier signal, Ac = 6 Frequency of message signal, fm = 300HzFrequency of carrier signal, fc = 11kHz The amplitude modulation index is given as, mi = Am/Ac = 40/6 > 1The modulated signal is given as,s=Ac*(1+mi*cos(2*pi*fm*t)).*cos(2*pi*fc*t);The plot of the signals can be seen below:From the plots, we can see that the signal, s(t), is distorted because the AM Index value is too high. Hence, the correct option is (a) The signal, s(t), is distorted because the AM Index value is too high.
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answer the next 3 questions using the following information: a project has three activities a, b, and c that must be carried out sequentially. the probability distributions of the times required to complete each of the activities a, b, and c are uniformly distributed in intervals [1,5], [2,3] and [3,6], respectively. find the total project completion time and run 1000 simulation trials in excel.
Answers
To find the total project completion time, we need to add the time required for each activity. Since the activities must be carried out sequentially, the total project completion time will be the sum of the times required for each activity.
The time required for activity a is uniformly distributed in the interval [1,5], so we can assume that the average time required for activity a is (1+5)/2 = 3.
The time required for activity b is uniformly distributed in the interval [2,3], so we can assume that the average time required for activity b is (2+3)/2 = 2.5.
The time required for activity c is uniformly distributed in the interval [3,6], so we can assume that the average time required for activity c is (3+6)/2 = 4.5.
Therefore, the total project completion time is:
Total project completion time = time for activity a + time for activity b + time for activity c
= 3 + 2.5 + 4.5
= 10
To run 1000 simulation trials in Excel, we can use the RAND function to generate random numbers between 0 and 1. We can then multiply these random numbers by the range of each activity to get a simulated completion time for each activity. We can then add up the simulated completion times for each trial to get a simulated total project completion time.
Here are the steps to run 1000 simulation trials in Excel:
1. In cell A1, enter "Trial" as the column header.
2. In cell B1, enter "Time for Activity A" as the column header.
3. In cell C1, enter "Time for Activity B" as the column header.
4. In cell D1, enter "Time for Activity C" as the column header.
5. In cell E1, enter "Total Project Completion Time" as the column header.
6. In cell A2, enter "1" as the first trial number.
7. In cell B2, enter the formula "=RAND()*(5-1)+1" to generate a random number between 1 and 5 for activity a.
8. In cell C2, enter the formula "=RAND()*(3-2)+2" to generate a random number between 2 and 3 for activity b.
9. In cell D2, enter the formula "=RAND()*(6-3)+3" to generate a random number between 3 and 6 for activity c.
10. In cell E2, enter the formula "=B2+C2+D2" to calculate the total project completion time for trial 1.
11. Select cells B2:E2 and drag the fill handle down to fill in the formulas for the remaining trials.
12. After filling in the formulas for 1000 trials, you can use Excel's AVERAGE function to find the average total project completion time across all trials.
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A cylindrical metal pipe has a diameter of 8.4 millimeters and a height of 10 millimeters. A hole cut out of the center has a diameter of 6 millimeters. A smaller cylinder is cut out of a larger cylinder. The smaller cylinder has a diameter of 6 millimeters. The larger cylinder has a diameter of 8.4 millimeters. Both cylinders have a height of 10 millimeters. What is the volume of metal in the pipe? Use 3.14 for and round the answer to the nearest tenth of a cubic millimeter. 282.6 mm3 271.3 mm3 553.9 mm3 836.5 mm3
Answers
Answer:
its b
Step-by-step explanation:
got it right on edge
The volume of the metal in the cylinderical pipe given the dimensions of the larger and smaller cylinders is 271.30 mm³.
What is the volume of the cylinder?A cylinder is a three-dimensional object. It is a prism with a circular base.
Volume of a cylinder = nr^2h
Where:
n = 22/7 r = radius = diameter / 2The volume of the metal in the cylinderical pipe = 3.14 x 10 x [(8.4/2)² - 6/2)²] = 271.30 mm³
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Suppose the population of a town is 17,400 and is growing 2% each year.
Predict the population after 9 years.
Answers
Answer: 20,532
Step-by-step explanation:
17,400(.02)=348
348(9)=3132
17,400+3132=20,532
Answer:
Step-by-step explanation:
This is an exponential growth function of the form:
\(y=ab^x\) where a is the initial population and b is the growth rate. This could also be written as
\(y=a(1+r)^x\) which is a bit more accurate. We'll use that one. r is the percent growth: 2% for us.
Filling in the formula:
\(y=17400(1+.02)^x\) and then simplifying:
\(y=17400(1.02)^x\)
That's the model for this population. Filling in any number of years for x will give you the population y. Our time is 9 years:
\(y=17400(1.02)^9\)
Take care of the exponents first, giving us:
\(y=17400(1.195092569)\) and then multiply to get
y = 20,794.6 or 20,794 people (since you can't have 6/10 of a person)
pls help me i can't figure this out and i have to finish it
Answers
Answer:
Because 20% of $10 will not be the same as 20% of a larger number
Step-by-step explanation:
120% of 10= 1.2(10)=$12
So we added $2 on the mark up.
Marking 20% off of $12 means the price will be 80% of 12= .8(12)= $9.60
This was a reduction of $2.40 which is 20% of $12
Answer:
Step-by-step explanation:
20% of 10 is $2
Marked up: 10+2 = $12
Marked down: 20% of 12 is 2.40 = 12-2.40 = $ 9.60
20% of 10 is not the same as 20% of 12.
PLS HELP ASAP rob is saving money for a new mp3 player. For every $15 he saves $8. On Sunday he earns $45 how much does he save
Answers
Answer:
the answer for this problem would be 24 dollars.
An object is traveling at steady speed of 9 4 /5 kilometer per hour how long will it take the object to travel 2 1/5 km first round the nearest integer to find the estimated answer then find the exact answer.
Answers
9.8km------------------------------------->1h
2.2km------------------------------------->xh
\(\begin{gathered} \text{Let's use cross multiplication:} \\ \frac{9.8}{2.2}=\frac{1}{x} \\ \text{solve for x:} \\ x=\frac{2.2}{9.8}=\frac{11}{49}h \\ x\approx13\min \\ x=\frac{660}{49}\min \end{gathered}\)sorry i chose the wrong image
Answers
Show calculations
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i just got some free money off of you kid
Can you please please please please please Please help me
Answers
Answer: 3 sqrt(21)
Step-by-step explanation:
wow this problem is gross!
making the triangle,
you need to use the law of cosines! searching "law of cosines" online should help a lot!
c^2 = a^2 + b^2 − 2ab cos(C)
you know C = 60 degrees (from 12 pm to 2pm!)
a = 12 cm and b = 15 cm
plug it into the equation and you get
c = 3 * sqrt(21) = 13.748
Answer: √70
Step-by-step explanation:
15^2=12^2 + x^2
225= 155 + x^2
X= √70
Consider the polynomials P1(t) = 2 + t + 3t2 + t3, P2(t) = 3+4+72 + 3t3, P3(t) = 1-3t+8t2 + 5t3, P4(t) = 5t + 5t2 + 3t3, Ps(t)--1+21+t2 + t3, which are all elements of the vector space Ps. We shall investigate the subspace W Span(pi(t), P2(t), Ps(t), pa(t), Ps(t) (a) Let v.-IA(t)le, the coordinate vector of P (t) relative to the basis ε-(Lt. fr Ps Enter (b) Let A be the matrix [vi v2 vs v4 vs]. Observe that Span(vi, v2, vs, v4, vs) -Col(A). Use these coordinate vectors into MATLAB as vi, v2, v3, v4, v5. this fact to compute a basis for Span[vi, V2, vs, V4, vs]. (Recall you can enter A into MATLAB as A-[vl v2 v3 v4 v5].) (c)Translate your previous answer into a basis for W (consisting of polynomials). What is dim W? (d) Is W- P3? Justify your answer
Answers
This gives us a basis for the subspace for all 3 parts where W of \(P_5,\)which is the column space of the matrix A.
(a) Let \(v_i\) be the coordinate vector of \(P_i\) relative to the basis \({P_1, P_2, P_3, P_4, P_5}.\) Then the matrix representation of A is:
A =\([v_1, v_2, v_3, v_4, v_5]\)
= [1 2 3 4 5]
[2 4 7 9 10]
[3 6 10 12 14]
[4 8 12 15 18]
[5 10 15 18 20]
Since Span \([v_i, v_2, v_s, v_4, v_s]\) is a subspace of \(P_5,\) its column space is a subspace of \(P_5\), which means Col(A) is contained in Span.
(b) Let A be the matrix \([v_1, v_2, v_3, v_4, v_5].\) We can use MATLAB to compute A as A = [1 2 3 4 5]. We can then use the basis vectors to compute a basis for Span by using the Gram-Schmidt process.
To do this, we first find a basis for Span\({v_i, v_2, v_s, v_4, v_s}:\)
\(v_i = [1 0 0 0 0]\\v_2 = [0 1 0 0 0]\\v_3 = [0 0 1 0 0]\\v_4 = [0 0 0 1 0]\\v_5 = [0 0 0 0 1]\)
Then we can compute the transformation matrix P from the basis\({v_i, v_2, v_3, v_4, v_5}\) to the standard basis {1, 2, 3, 4, 5}:
P = [1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
Finally, we can use the transformation matrix P to find a basis for the subspace Span \({v_i, v_2, v_s, v_4, v_s}:\)
P = [1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[0 0 0 0 0]
[0 0 0 0 0]
This gives us a basis for the subspace Span \({v_i, v_2, v_s, v_4, v_s}\) of P_5, which is the column space of A.
(c) To find a basis for the subspace W of \(P_5,\) we can use the same method as in part (b). The basis vectors of W are the polynomials in \(P_5\)that are in the span of the polynomials in \({P_1, P_2, P_3, P_4, P_5}.\)
Since \(P_1, P_2, P_3, P_4, P_5\) are linearly independent, the polynomials in their span are also linearly independent, so W is a proper subspace of P_5.
To find a basis for W, we can use the Gram-Schmidt process as before, starting with the standard basis vectors {1, 2, 3, 4, 5}:
\(v_i = [1 0 0 0 0]\\v_2 = [0 1 0 0 0]\\v_3 = [0 0 1 0 0]\\v_4 = [0 0 0 1 0]\\v_5 = [0 0 0 0 1]\)
Then we can compute the transformation matrix P from the basis \({v_i, v_2, v_3, v_4, v_5}\) to the standard basis {1, 2, 3, 4, 5}:
P = [1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
Finally, we can use the transformation matrix P to find a basis for the subspace W:
P = [1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[0 0 0 0 0]
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What is the fractional equivalent of the repeating decimal n = 0.1515... ? answer the questions to find out. 1. how many repeating digits does the number represented by n have? (2 points)
Answers
The fractional equivalent of the repeating decimal would be 15/99 and 0.1515 ... is a repeating decimal so it does n't end , but in the shown number is 4 digits .
The fractions with distinct numerators and denominators but the same value are said to be equivalent fractions. For instance, since 2/4 and 3/6 both equal the 12, they are identical fractions. An element of a whole is a fraction.
Multiply a fraction's numerator and denominator by the same number of digits. You won't alter the value of the fraction and will produce an identical fraction if you multiply the top and bottom of the fraction by the same number.
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