Sunday

December 21, 2014

December 21, 2014

Total # Posts: 28,261

**AP Calc**

f ' (x) = x(1/x) + lnx + (1-x)(-1/(1-x)) + -1(ln(1-x) = 1 + lnx -1 - ln(1-x) = lnx - ln(1-x) = ln(x/(1-x)) = 0 for a min then e^0 = x/(1-x) 1 = x/(1-x) 1-x = x x = 1/2 Proof: confirmed by Wolfram http://www.wolframalpha.com/input/?i=plot+y+%3D+x+ln+x%2B%281-x%29ln%281-x%29
*December 20, 2014*

**AP Calc**

I believe this is the same question, and answered by Steve http://www.jiskha.com/display.cgi?id=1419047350
*December 20, 2014*

**Math - Two-Step Inequalities - Please Help**

–3 > –3t + 6 3t > 6+3 3t > 9 t > 3 So for the given numbers, I see that only 4 > 3 so , yes, C is the correct answer.
*December 19, 2014*

**Math**

You have to look for some pattern Did you notice that for every 1 unit increase in the x's there is a 5 unit increase in the y's so you know it starts with y = 5x + b, we still don't know the b plug in one of the points , how about (1,2) 2 = 5(1) + b b = -3 ...
*December 19, 2014*

**math**

"Using 22/7 for pi is just not rational! " good one, they were selling t-shirts with that slogan to the math-nerds at our University of Waterloo a few years back.
*December 19, 2014*

**math**

I think 22/7 is the approximation we used for π in the "olden days" so a + b = π/2 or a and b are complementary angles Construct a right angled triangles with angles a and b along with the right angle since sina = 1/3, the missing side is √8 by ...
*December 19, 2014*

**Algebra**

x • b • y • a = (xy)(ab) = (1/37)(37) = 1
*December 18, 2014*

**math**

Elimination is a poor choice for this question. I would definitely use substitution. sub the 1st into the 2nd 2(5y) - 3y = 7 10y-3y=7 7y=7 y = 1 , then x = 5 compare this with elimination: 5y = x x - 5y = 0 multiply by 2, 2x - 10y = 0 2x - 3y = 7 subtract them -7y = -7 y = 1 ...
*December 18, 2014*

**Algebra math**

In English please.
*December 18, 2014*

**math**

number of people ---- n Company A: CostA = 25n + 50 Company B: CostB = 20n + 130 So for n = 100 CostA = 2500+50 = 2550 CostB = 2000+130 = 2130 When is CostA = CostB ? 25n + 50 = 20n + 130 5n = 80 n = 16 break-even number is 16 people.
*December 18, 2014*

**Math**

Why not just reverse the process and subtract 2 from top and bottom to get 2/3 ?
*December 18, 2014*

**Calculus**

done, look back please
*December 18, 2014*

**math**

first one is correct #2, 6.3% of $77 = .063(77) = $4.85 77+4.85 = 81.85 , so #4
*December 18, 2014*

**Calculus**

First of all you must understand what the intersection is When you intersect two planes you get a straight line, unless the two planes are parallel In this case they are not, since 1:1:-1 ≠ 2:4:-3 multiply the first by 3 3x+3x-3z = -36 subtract 2x+ 4y - 3z = -8...
*December 18, 2014*

**Math**

The inclusive vs exclusive restrictions in rational expressions always leads to debate. I restricted x = -1 and x = -2 since subbing in these values in the original yields 0/0 but subbing those values in my answer (and yours) yields a real number Since I was using the equal ...
*December 18, 2014*

**Math**

I read that as: (x^2 - 2x)/(x+1) * (x^2 - 1)/(x^2 + x - 6) =x(x-2)/(x+1)*(x-1)(x+1)/( (x+3)(x-2) = x(x-1)/(x+3) or (x^2 - x)/(x+3) , x ≠ -1,2
*December 18, 2014*

**math**

Please number your question, how are we to know who is who and what you want done 1. assuming you want to solve 2/3n = 8 multiply each term by 3 2n = 24 n = 12 2. sub the 2nd into the 1st 4x + 3y = 13 4x + 3(-x+4) = 13 4x - 3x + 12 = 13 x = 1 back into the 2nd, y = -1+4 = 3 x=...
*December 18, 2014*

**Algebra**

good job
*December 18, 2014*

**Algebra**

first get rid of nasty fractions by multiplying each term by 3 , LCD 3x^2 - 5x + 2 = 0 which factors quite nicely. Let me know what you got
*December 18, 2014*

**algebra**

speed of boat --- x speed of current -- y x+y = 21 x-y = 5 add the two equations, and continue
*December 18, 2014*

**Algebra**

You typed the equation incorrectly, it should say y = 5 (1/2)^(t/20) so ... 4 = 5(1/2)^(t/20) .8 = (.5)^(t/20) take log of both sides log.8 = log(.5^(t/20)) log.8 = (t/20)log.5 log.8/log.5 = t/20 t = 20(log.8)/log.5) I leave the button-pushing up to you
*December 18, 2014*

**stats**

a) prob = (25/32)(24/31) = .. b) prob = (25/32)(7/31) = ... c) prob (both red) = answer to a) prob(both black) = (7/32)(6/31) = .. prob(different colour) = 1 - a) - both black
*December 18, 2014*

**math algebra**

Let the length along the front be x let the width be y let the height be z x^2+ z^2 = 25^2 = 625 , #1 y^2 + z^2 = 20^2 = 400 , #2 subtract them x^2 - y^2 = 225 , #3 also x^2 + y^2 + z^2 = 28^2 = 784 , #4 add it to #3 2x^2 + z^2 = 1009, but x^2 + z^2 = 625 from #1 so 2x^2 + z^2...
*December 18, 2014*

**math algebra**

recall that if the line is written in the form y = mx + b, then m is the slope so ... 8y = -16 or, how about 8y = 0x - 16 y = (0/8)x - 2 y = 0x - 2 so the slope must be 0 in general if the linear equation can be written in the form y = k, where k is just a constant, you have a...
*December 18, 2014*

**Math**

Zyere is not correct 72/60 = x/40 60x = 40(72) 60x = 2880 x = 48 The ratio of height : shadow must be the same note: 72 : 60 = 6 : 5 , and 48 : 40 = 6 : 5
*December 18, 2014*

**Math**

use a simple ratio and solve for x 72/60 = x/40
*December 18, 2014*

**math algebra**

Use the last two points to find the equation slope = (697-522)/(100-75) = 175/25 = 7 so we have y = 7x + b sub in (75,522) 522 = 7(75) + b b = -3 so we have y = 7x - 3 From your knowledge of y = mx + b, what do you think?
*December 18, 2014*

**math**

Let the capacity of the tank be x L (1/4)x + 90 = (3/4)x times 4 x + 360 = 3x 2x = 360 x = 180
*December 18, 2014*

***repost* - Grade 11 math**

sketch a right-angled triangle. On the horizontal base mark W for Wes. let the fireworks be at P and let Q be the point below P, so that angle Q = 90° let the angle at W be 28° let A for Abby be a point between W and Q so that angle PAQ = 40° , and AW = 20 Look at ...
*December 17, 2014*

**Algebra 1**

http://www.wolframalpha.com/input/?i=plot+y+%3D+%7Cx%7C+%2C+y+%3D+%7Cx+%2B2%7C
*December 17, 2014*

**math**

ar^2 = 25 ar^7 = 25/32 divide the 2nd by the 1st: r^5 = 25/32 ÷ 25 = 25/800 = 1/32 r = 1/2 in ar^2 = 25 (1/4)a = 25 a = 100 where does t come into the picture ?
*December 17, 2014*

**Math (Repost)**

1. u^-3 = 1/125 1/u^3 = 1/125 u^3 = 125 u = 5 2. y-1 = 4y - 2/3 each term times 3 3y - 3 = 12y - 2 -9y = 1 y = -1/9 3. your are right 4. number of miles last week --- x x+40 = 4x + 10 -3x = -30 x = 10 She ran 10 miles last week 5. change it into the form y = mx + b (takes 2 ...
*December 17, 2014*

**Algebra 2**

= 200(1 + .08/12)^12 = 200(1 + .0066666..)^12 = 200(1.006666...)^12 = appr 216.60
*December 17, 2014*

**Algebra 1**

slope of given line = 8/3 so slope of your new line = -3/8 through (-2,3) y = 3 = (-3/8)(x+2) since the original was written in that form, I assume the new one is to be in that same format.
*December 17, 2014*

**Math**

(m + s)/2 = 74 m+s = 148 (m + e )/2 = 83 m+ e = 166 subtract them: e - s = 18
*December 17, 2014*

**Math**

Did you notice that if you multiply the 2nd equation by 4, you get the second? So , you basically only have one equation, thus an infinite number of solutions.
*December 17, 2014*

**Algebra 2**

you know that it must be (x+y)^9 to get a ?x^7y^2 so the term is C(9,7) x^7 y^2 = 36x^7y^2 Observe that the 10th row of Pascal's triangle is 1 9 36 84 126 126 84 36 9 1
*December 17, 2014*

**Algebra**

subtract your equations: 1 Sandwich = 4.75 in the 1st: 4.75 + Salad = 8.7 salad = 3.95
*December 17, 2014*

**MATH PLEASE HELP**

1 e^.012t = 2 take ln of both sides ln e^.01t = ln2 .012t ln e = ln 2 , but ln e = 1 .12t = ln 2 t = ln2/.012 = 57.76 years or 58 years to the nearest year.
*December 17, 2014*

**Math Help**

5^x =125 <----> log5 125 = x I had my students memorize this pattern, 2^3 = 8 <---> log28 = 3
*December 17, 2014*

**Algebra**

total area = 12000 ft^2 still to be mowed = (1/2)(12000) = 8000 ft^2 let the width mowed by Lisa be x ft wide area still to be done = (100-2x)(120-2x) = 12000-200x - 240x + 4x^2 4x^2 - 440x + 12000 = 8000 x^2 - 110x + 1000 = 0 (x-100)(x -10) = 0 x = 100 or x = 10 , but clearly...
*December 17, 2014*

**Calculus**

Steve , you are right, I read the question rather superficially. (notice it was my last post last night, must have been a bit tired)
*December 16, 2014*

**Calculus**

f(x) = e^(4x^2) f(-1/4) = e^(1/4) f(1/4) = e^(1/4) avg value = (e^(1/4) - e^(1/4) )/(1/2) = 0
*December 16, 2014*

**math help!!**

This one is easier than the previous question you posted about GS's Give this one a try, you can do it.
*December 16, 2014*

**math please hellp**

Done, look back
*December 16, 2014*

**algebra 1**

when it hits the water, h = 0 0 = -16t^2 + 20t + 24 4t^2 - 5t - 6 = 0 (t-2)(4t + 3) = 0 t = 2 or t = -3/4, the last answer is inadmissible It will take 2 seconds I have no idea what the last part of your post is. Is that supposed to be a solution ? If so, it makes no sense at ...
*December 16, 2014*

**math help**

a + ar = 12 a(1+r) = 12 , #1 a + ar + ar^2 = 62 a(1 + r + r^2) = 62 , #2 divide #2 by #1 (1+r+ r^2)/(1+r) = 62/12 = 31/6 6r^2 + 6r + 6 = 31 + 31r 6r^2 - 25r - 25 = 0 (r-5)(6r + 5) = 0 r = 5 or r = -5/6 if r = 5 , in #1 6a = 12 a = 2 the first 4 terms are 2 , 10, 50, 250 if r...
*December 16, 2014*

**math**

Draw a right-angled triangle DAC, AC as a base line Let CD be the height of the building. Let B be a point between A and C, so that AB = 230 angle A = 28° angle DBC = 43° Look at triangle ABD, you know angle A = 28, angle ABD = 180-43 = 137 angle ADB = 15° you can ...
*December 16, 2014*

**Math**

Just did this when you posted it as "John"
*December 16, 2014*

**calculus**

Just like taking slope C(101) = 101^2 - 10000 = 201 C(100) = 100^2 - 10000 = 0 average rate of change = (201-0)/(101-100) = 201 instantaneous rate of change = 2x at x = 100 rate = 200
*December 16, 2014*

**Algebra 1**

y+7 = -.54(x-4)
*December 16, 2014*

**math**

double the first: 2x + 4y = 16 subtract from the 2nd x = 4 back into the 1st: 4 + 2y = 8 2y = 4 y = 2 x = 4, y = 2
*December 16, 2014*

**algebra**

prob = (10/24(10/24) = 25/144
*December 16, 2014*

**Math**

1st + last = 100 2nd + 2nd last = 100 ... ... looks like I can form 50 such pairs sum = 50(100) = 5000 use the same concept for the second question.
*December 16, 2014*

**Literature - the Watch**

I am sure that you were not assigned this yesterday. In my days we used to call this "planning ahead". Giving you a summary would be contributing to plagiarism . Here is a page that lets you read the short story on line: http://www.readbookonline.net/readOnLine/22270/
*December 16, 2014*

**MATH**

When I divide 56 by 3 , I get 18 packages with 2 brownies left over. So none of your choices fit.
*December 16, 2014*

**math**

assuming a and b are constants, we have (1/a^n) x^n + (1/b^n) y^n = 2 (n/a^n) x^(n-1) + (n/b^n) y^(n-1) dy/dx = 0 so for (a,b) (n/a^n)(a^(n-1) + (n/b^n)(b^(n-1) dy/dx = 0 (n/a) + (n/b) dy/dx = 0 dy/dx = -(n/a) (b/n) = -b/a equation of tangent at (a,b) y - b = (-b/a)(x-a) toy ...
*December 16, 2014*

**one math question.**

yes
*December 16, 2014*

**contains typo - ^Maths^**

You have a typo. the right side of your equation should say 72^y log3(x-7) = 2y --> 3^(2y) = x-7 log2(x-6) = 3y --> 2^(3y) = x-6 then LS = x^2 - 13x + 42 = (x-6)(x-7) = ( 2^(3y) ) ( 3^(2y) ) = (8^y)(9^y) = 72^y = RS
*December 16, 2014*

**math algebra**

Obviously the diagonal of 10 must be greater than either the length, width or height, so the height cannot be greater than 10 and you are incorrect Look at the diagonal of the base, call it d d^2 = 8^2 + 2^2 = 68 d = √68 now the diagonal we want is the hypotenuse of the ...
*December 15, 2014*

**algebra**

so his effective speed downriver is 4+3 or 7 mph and he goes for 5 hrs distance = rate x time = ....
*December 15, 2014*

**Math**

A GS, with a = -4 r = -1/4 sum(∞) = a/(1-r) = -4/(1 -(-1/4) ) = -4/(5/4) = -16/5 or -3.2
*December 15, 2014*

**MATH**

= .5^0 + .5^1 + .5^2 + .5^3 + .5^4 = 1 + .5 + .5^2 + .5^3 + .5^4 or = 1 + 1/2 + (1/2)^2 + (1/2)^3 + (1/2)^4 we could just add them up, but I guess this question deals with the sum of a GS a = 1 r = .5 n = 5 sum(5) = 1(.5^5 - 1)/(.5 - 1) = (.03125 - 1)/(-.5) = 1.9375 or 31/16
*December 15, 2014*

**Math**

= 2/2 + 3/2 + 4/2 + ... 8/2 = ∑ (1+n)/2 for n = 1 to 7 This answer is not unique, there are many ways to express it
*December 15, 2014*

**Math**

a+(a+1)+(a+2)+.....+(a+n-1) looks like an arithmetic series with first term a common difference = a+1 - a = 1 for n terms so sum = (n/2) (2a + (n-1)(1)) = (n/2)(2a + n-1) so (n/2)(2a + n - 1) = 100 2an + n^2 - n = 200 2an = 200 - n^2 + n a = (200 - n^2 + n)/(2n) n .. a 2 - 49....
*December 15, 2014*

**Algebra**

Tell me what your answer is and I will critique it.
*December 15, 2014*

**Algebra**

The first element of the vertex in this case would be the time it took to reach the highest point, and the 2nd number would be that maximum height.
*December 15, 2014*

**Algebra**

the x-intercept would be the time taken to hit the ground
*December 15, 2014*

**Math**

Memorize the relationship: π radians = 180° from that you can solve for each unit : 1 radian = (180/π) ° <-------> 1 ° = (π/80) radians so which ever is given, proceed from there (13π/12) rad = (180/π)(13π/12) ° = 195° (...
*December 15, 2014*

**math**

one hectare is 10,000 m^2 , (100 m by 100m) so half of that is 5,000 m^2 so let each side be x then x^2 = 5000 for the diagonal d, d^2 = x^2 + x^2 = 2x^2 = 10000 d = 100 m
*December 15, 2014*

**7th grade math**

#1, at 10^2 - 1 = 100-1 = 99 #14, leave your answer as an inequality, since that is what you started with, thus c < -12.8 the rest are correct
*December 15, 2014*

**Calculus (math)**

if v(t) = t^3 - 5t^2 s(t) = (1/4)t^4 - (5/3)t^3 + c s(6) = (1/4)6^4 - (5/3)6^3 = -36 + c s(-1) = (1/4)(-1)^4 - (5/3)(-1)^3 = 23/12 + c distance traveled = 23/12+c - (-36+c) = 4550/12
*December 14, 2014*

**Steve saw the light - easy geometry**

The fact that the sides were not listed sequentially made me interpret the question in an entirely different way, so go with Steve' solution. Anyway, my interpretation led to a nice problem, lol
*December 14, 2014*

**For #1 easy geometry**

From my previous reply to this problem http://www.jiskha.com/display.cgi?id=1418471605 (it was called #3 ) I had given you "I drew a figure ABCD , with AD || BC AD = 10 AB = 20 BC = 30, and CD = 10 From A draw a line AP, so that AP is || to DC and P is on BC So AP=10 and ...
*December 14, 2014*

**easy geometry**

I'll take care of #2 first. the common formula is A = P(1 + r)^n , where r is the rate per interest period expressed as a decimal (e.g. 3% --> .03 ) and n is the number of interest periods I don't know where your a comes from. so 1690 = 1000(1 + r)^2 (1+r)^2 = 1.69 ...
*December 14, 2014*

**math**

i = .09/4 = .0225 n = 4(8) = 32 payment = P P( 1.0225^32 - 1)/.0225 = 8000 solve for P
*December 14, 2014*

**precalculus**

for domain, all you have to worry about is the denominator and make it does not become zero. x^2 + x + 9 = 0 has no real solution, (the b^2-4ac is negative) so the domain is the set of all real numbers for x b) Thus, since there are no breaks in the function, it is continuous ...
*December 14, 2014*

**Calculus 1**

I drew a circle and placed the woman at point A. She wants to go to point B, clearly where AB is a diameter. Label the centre as O. Pick a point P on the circle so that she will row from A to P, and then walk along the circumference from P to B I labeled AP = x and arc PB as a...
*December 13, 2014*

**Algebra**

from the 2nd: 4x = 20 - 3y x = 5 - (3/4)y plug into the 1st: 5x - 7y = -18 5(5 - (3/4)y) - 7y = -18 25 - 15y/4 - 7y = -18 -43y/4 = -43 y = -43(-4/43) = 4 x = 5 - (3/4)4 = 2 (substitution would have been my last choice in the methods to solve this, I would have used elimination)
*December 13, 2014*

**AP CaLC. AB**

Here is a solution to your problem, with sides 6 and 12 I suggest you print out the solution and change the constants to yours. I suggest you are careful that your base of x, and height of y are consistent with the x and y in the solution https://www.math.ucdavis.edu/~kouba/...
*December 13, 2014*

**AP CaLC. AB**

Where do x and y enter the picture ??
*December 13, 2014*

**Algebra**

strange question, clearly 9x^2 - 12x + 4 = 49 (3x - 2)^2 = 49 3x - 2 = ± 7 3x = 2 ± 7 3x = 9 or 3x = -5 x = 3 or x = -5/3 so I guess they just wanted the 2nd line .
*December 13, 2014*

**math**

This implies that either 4, 5 or all 6 will pass prob = C(6,4) (.6)^4 (.4)^2 + C(6,5) (.6)^5 (.4) + C(6,6) .6^6 I get .54432
*December 13, 2014*

**math**

arggh, go with Damon's factoring of the last part Whatever made me think that 2x5 = 16 ?????? It's clearly 2 x 8
*December 13, 2014*

**math**

your expression definitely needs brackets to make sense (n^2-4)/(n^2-2n) * n/(n^2+10n+16) = (n+2)(n-2)/(n(n-2) ) * n( (n+2)(n+5) = 1/(n+5) , n ≠ -2 , 0, 2 , -5
*December 13, 2014*

**math**

another way: find the intersection line of the two planes add the two equations, 4x -3y = 10 y = (4x-10)/3 pick any "nice" value of x let x = 1, y = -2 in the 2nd equation: 3 + 2 + z = 4 , z = --1 --> point (1,-2,-1) let x = 10, y = 10, in the 2nd 30 - 10 + z = 4...
*December 13, 2014*

**math**

year 1 = 19000 year 2 = 19000(1.04) year 3 = 19000(1.04)^2 .. year 10 = 19000(1.04)^9 I somehow expect that "they" want you to add them up. It would be a GS, with a = 19000, r = 1.04 , n = 10 BUT, from an actuarial point of few monies cannot just be added up unless ...
*December 13, 2014*

**Algebra 1**

y = |x-2| why not just take some simple values of x and see what points you get, then plot them 0,2 1,1 2,0 3,1 4,2 ... -1,3 -2,4 -3,5 .. can you see the pattern ?
*December 13, 2014*

**Algebra**

t2 = a + d = 2 t9 = a + 8d = 30 subtract them 7x = 28 d = 4 in a+d=2 a + 4 = 2 a = -2 t15 = a + 14d = -2 + 14(4) = 54
*December 13, 2014*

**#4 easy geometry**

x-17 + 24-x + 3x-40 + x+1 = 42 4x = 74 x = 18.5 , so the sides are 1.5 , 5.5, 15.5 , and 19.5 how about doubling each side, creating a "similar" quadrilateral, (angles won't change) 3 , 11 , 31 , and 39 , mmmh, what do you know about sides of a parallelogram ? #5...
*December 13, 2014*

**easy geometry**

1. In a rhombus, the diagonals bisect each other at right angles. So you have 4 congruent right-angled triangles with sides 5 and 12, and the hypotenuse is the side of the rhombus. s^2 = 5^2 + 12^2 (You should recognize that Pythagorean triple, if not, remember it) Find the ...
*December 13, 2014*

**Probability**

To get a plain cookie: we could have (1,P) or (2,P) prob (1, P)= (1/2)((30/40) = 3/8 prob (2,P) = (1/2)(20/40) = 1/4 prob(plain cookie) = 3/8 + 1/4 = 5/8 (checking: prob(1,C) = (1/2)(10/40) = 1/8 prob(2,C) = (1/2)(20/40) = 1/4 note 3/8+1/4+1/8+1/= 1 , check! ) could have ...
*December 12, 2014*

**Math**

PV = 5 e^-.03 + 5 e^-.06 + ... + 5 e^-(10(.03)) I see a GS with a = 5 e^-.03 r = e^-.03 n = 10 PV = a(r^n - 1)/(r-1) 5 e^-.03( e^-.3 - 1)/(e^-.03 - 1) = appr 43.85 million dollars
*December 12, 2014*

**College math 100**

white ribbons --- x blues ---------- x-2 reds ----------- x+3 form the equation and continue
*December 12, 2014*

**Math**

5 Σ 4(3/4)^k k=0 = 4(3/4)^0 + 4(3/4)^1 + 4(3/4)^2 + 4(3/4)^3 + 4(3/4)^4 + 4(3/4)^5 = 4[ 1 + (3/4) + (3/4)^2 + (3/4)^3 + (3/4)^4 + (3/4)^5 ] The part in the brackets is a GS with a = 1 , r = 3/4 and n = 6 = 4 ( 1( (3/4)^6 - 1)/(3/4 - 1) =4 (729/4096 - 1)/(-1/4) = -16(-3367...
*December 12, 2014*

**Math**

no 1/2 * 6 + 9^2 - (18/6) = 3 + 81 - 3 = 81
*December 11, 2014*

**math**

In a true scale , the units of the terms are the same, and thus the units would cancel 8 cm : 1000 m = 8 cm : 100000 cm = 1 : 12500 behold the beauty and simplicity of the metric system Had this been 1000 yards and 8 inches, it would be a lot messier.
*December 11, 2014*

**algebra**

Where did you get a kitten to weigh 1.5 kg ?? 4x+4x= 10 8x=10 x = 1.25 kg so each kitten weighs 1.25 kg and the cat weighs 4x or 4(1.25) = 5 kg check: does the cat weigh 4 times as much as a kitten what is 4(1.25) ? = 5, check! what is the total weight of the 4 kitten and the ...
*December 11, 2014*

**algebra**

kitten --- x kg cat ------4x kg 4(x) + 4x = 10 take over
*December 11, 2014*

**math**

It asked for the ratio which cannot just be one number. Ratios express the relationship between 2 or more items so 180:20 = 9 : 1
*December 11, 2014*

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