Two positive numbers are in the ratio 2 : 3. Find the numbers if ​(a) their sum is 55 and ​(b) their product is 384.

Answers

Answer 1

say the numbers are "a" and "b", we know that they're in the ratio of 2:3, or namely that

[tex]\stackrel{\textit{in the ratio of 2 to 3}}{a~:~b\qquad \qquad 2:3}\qquad \qquad \implies \qquad \cfrac{a}{b}=\cfrac{2}{3}\implies \cfrac{3a}{2}=b \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{their sum is 55}}{a + b = 55}\implies a + \cfrac{3a}{2}=55\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{2}}{2\left( a + \cfrac{3a}{2} \right)=2(55)} \\\\\\ 2a+3a=110\implies 5a=110\implies a = \cfrac{110}{2}\implies \boxed{a = 22}[/tex]

[tex]\stackrel{\textit{we know that}}{\cfrac{3a}{2}=b}\implies \cfrac{3(22)}{2}=b\implies \boxed{33=b} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{their product is 384}}{ab=384}\implies a\left( \cfrac{3a}{2} \right)=384\implies \cfrac{3a^2}{2}=384\implies 3a^2=768 \\\\\\ a^2=\cfrac{768}{3}\implies a^2=256\implies a=\sqrt{256}\implies \boxed{a=16} \\\\\\ \stackrel{\textit{we know that}}{\cfrac{3a}{2}=b}\implies \cfrac{3(16)}{2}=b\implies \boxed{24=b}[/tex]

Answer 2

The positive numbers for both the parts are obtained as 22,33 and 16,24 respectively.

How to write a linear equation?

A linear equation for the given case can be written by assuming any variable as the unknown quantity. Then, as per the given data the required operations are done and it is equated to some value.

(a) As per the question, the numbers can be considered as 2x and 3x respectively.

Then, the equation for their sum can be written as follows,

3x + 2x = 55

⇒ 5x = 55

⇒ x = 11

The numbers are obtained as 2 × 11 = 22 and 3 × 11 = 33.

(b) Similarly, the equation for the product of numbers can be written as,

3x × 2x = 384

⇒ 6x² = 384

⇒ x² = 64

⇒ x = 8

The numbers are obtained as 2× 8 = 16 and 3 × 8 = 24.

The required number for both the cases are 22,33 and 16,24 respectively.

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Related Questions

what does 5/6+1/3 equal

Answers

Answer:

5/6 + 1/3 = 7/6

Step-by-step explanation:

Let f(x) = x+2/x+6
f^-1(-6) =

Answers

We can explicitly find the inverse. If [tex]f^{-1}(x)[/tex] is the inverse of [tex]f(x)[/tex], then

[tex]f\left(f^{-1}(x)\right) = \dfrac{f^{-1}(x)+2}{f^{-1}(x)+6} = x[/tex]

Solve for the inverse :

[tex]\dfrac{f^{-1}(x) + 2}{f^{-1}(x) + 6} = x[/tex]

[tex]\dfrac{f^{-1}(x) + 6 - 4}{f^{-1}(x) + 6} = x[/tex]

[tex]1 - \dfrac4{f^{-1}(x) + 6} = x[/tex]

[tex]1 - x = \dfrac4{f^{-1}(x) + 6}[/tex]

[tex]f^{-1}(x) + 6 = \dfrac4{1-x}[/tex]

[tex]\implies f^{-1}(x) = \dfrac4{1-x} - 6[/tex]

Then when x = -6, we have

[tex]f^{-1}(-6) = \dfrac4{1-(-6)} - 6 = \dfrac47-6 = \boxed{-\dfrac{38}7}[/tex]

Alternatively, we can first solve for x such that [tex]f(x) = -6[/tex]. Then taking the inverse of both sides, [tex]x = f^{-1}(-6)[/tex]. (The difference in this method is that we don't compute the inverse for all x.)

We have

[tex]\dfrac{x+2}{x+6} = -6[/tex]

[tex]x + 2 = -6 (x + 6)[/tex]

[tex]x + 2 = -6x - 36[/tex]

[tex]7x = -38[/tex]

[tex]\implies x = \boxed{-\dfrac{38}7}[/tex]

A = $2100, P = $1600, t = 8

r = %

Answers

Answer:

I want to say 900 many sorry if it's wrong

Help me I need the math problem

Answers

Answer:

y=4x-3

Step-by-step explanation:

Answer:

DO i care... no ask your dam teacher won der why u aint pass

Step-by-step explanation:

9
11 (a) Grapes cost £2 per kilogram.
Calculate the cost of 380g of grapes.

Answers

Answer:

£0.76

Step-by-step explanation:

1kilogram = 1000grams

1000g of grapes costs £2

380g costs = (380 × 2)/ 1000

760/1000 = £0.76

point w is graphed at (n,1)

Answers

Answer:

find the vertical asymptote of the graph of f(×) = (×+3) +2.

Examine the figure. What is the measure of angle a?

Answers

Answer:

59

Step-by-step explanation:

90+31=121

180-121=59

Answer:

a=59

basic fact:

angles on a straight line add up to 180

Step-by-step explanation:

we can ignore b for a start

31+90+a=180

121+a=180

121+59=180

a=59

Which expression is equivalent to 1/4 (5x + 6)?

Answers

The expression which is equivalent to 1/4 (5x + 6) is; Choice A: {5(1/4)x} + {6(1/4)x}.

According to the question:

We are required to determine an expression which is equivalent to 1/4 (5x + 6).

In a bid to expand the expression; we must multiply each term in the parenthesis by (1/4);

In essence; we have;

{5(1/4)x} + {6(1/4)x}

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2. The sum of two cubes can be factored by using the formula q3 + b3 = (a + b)(a? - ab + b).
(a) Verify the formula by multiplying the right side of the equation.
(b) Factor the expression 8x3 + 27.
(c) One of the factors of 23 - bºis a - b. Find a quadratic factor of a3 - bº Show your work.
(d) Factor the expression x3 - 1.

Answers

Factorization involves breaking down an expression.

The factorized expression of [tex]8x^3 + 27[/tex] is [tex]8x^3 + 27 =(2x + 3)(4x^2 -8x3 +9)[/tex]The other factor of [tex]2^3 - b^3[/tex] is [tex]4+2b+b^2[/tex]The factorized expression of x^3 - 1 is [tex]x^3 - 1^3=(x-1)(x^2+x+1)[/tex]

The sum of cubes is given as:

[tex](a^3 + b^3) = (a + b)(a^2 -ab + b^2)[/tex]

(a) Verify the formula

Expand the expression on the right-hand side

[tex](a^3 + b^3) = a^3 -a^2b + ab^2 +a^2b - ab^2 + b^3[/tex]

Collect like terms

[tex](a^3 + b^3) = a^3 -a^2b +a^2b+ ab^2 - ab^2 + b^3[/tex]

[tex](a^3 + b^3) = a^3 + b^3[/tex]

The formula has been verified

(b) Factorized 8x^3+ 27

We have:

[tex]8x^3 + 27[/tex]

Express 27 as 3^3

[tex]8x^3 + 27 =8x^3 + 3^2[/tex]

Express 8 as 2^3

[tex]8x^3 + 27 =2^3x^3 + 3^3[/tex]

Rewrite as:

[tex]8x^3 + 27 =(2x)^3 + 3^3[/tex]

Given that:

[tex](a^3 + b^3) = (a + b)(a^2 -ab + b^2)[/tex]

The expression becomes

[tex]8x^3 + 27 =(2x + 3)((2x)^2 - (2x)3 +3^2)[/tex]

[tex]8x^3 + 27 =(2x + 3)(4x^2 -8x3 +9)[/tex]

(c)The other factor of 2^3 - b^3

By difference of cubes, we have:

[tex]x^3 - y^3=(x-y)(x^2+xy+y^2)[/tex]

So, the equation becomes

[tex]2^3 - b^3=(2-b)(2^2+2b+b^2)[/tex]

This gives

[tex]2^3 - b^3=(2-b)(4+2b+b^2)[/tex]

Hence, the other factor of [tex]2^3 - b^3[/tex] is [tex]4+2b+b^2[/tex]

(c) Factor x^3 - 1

We have:

[tex]x^3 - y^3=(x-y)(x^2+xy+y^2)[/tex]

Express 1 as 1^3 in x^3 - 1

[tex]x^3 - 1 =x^3 - 1^3[/tex]

[tex]x^3 - y^3=(x-y)(x^2+xy+y^2)[/tex] becomes

[tex]x^3 - 1^3=(x-1)(x^2+x+1^2)[/tex]

[tex]x^3 - 1^3=(x-1)(x^2+x+1)[/tex]

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A passenger plane flew from Texas to Frankfurt and back. The round trip is 9900 km.
return trip took 4 hours less than the trip to Frankfurt because of the 50 km/hr wind over the ocean on the return trip. Assume there was no wind on the trip to Frankfurt. Let x km/hr represent the average speed of the plane in still air.

Answers

We want to find the average speed of the plane in still air, we will get that the average speed of the plane in still air is 355.3 km/hr

So we know that the total trip is 9900km, so from Texas to Frankfurt the distance is half of that:

9900km/2 = 4950km

Now, The speed of the plane in still air is x km/hr

The speed of the air is 50 km/hr, and it flows from Frankfurt to Texas.

So in the first part of the trip, the wind blows in opposite direction to the motion of the plane, thus the speed of the plane will be:

(x - 50) km/hr

While in the return trip the wind blows in favor of the plane, in this case, the speed is:

(x + 50)km/hr

Then if the first part of the trip lasts for T hours (the second for T - 4 hours) we can write the system of equations:

(x - 50)*T = 4950

(x + 50)*(T - 4) = 4950

Where I removed the units so it is easier to read.

If we isolate T in the first equation we get:

T = 4950/(x - 50)

Now we can replace this in the other equation to get:

(x + 50)*(4950/(x - 50)  - 4) = 4950

If we multiply both sides by (x - 50) we get:

(x + 50)*4950 - 4*(x + 50)*(x - 50) = 4950*(x - 50)

Now we can solve this for x:

(x + 50)*4950 - 4*(x^2 - 2500) - 4950*(x - 50) = 0

100*4950 - 4*(x^2 - 2500)  = 0

495000 - 4*x^2 + 10000 = 0

505000 = 4*x^2

√(505000/4) = x = 355.3

The average speed of the plane in still air is 355.3 km/hr

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I need help fast I forgot how to do this

Answers

Answer:

ABC & E are true statements about the graph

5. What is the measure of Angle WVX in the given square?

Answers

Answer:

try 90°C for an answer, good luck

last month, a store sent 2,014 emails to customers about sales. the number of e-mails sent the month before was 2,104. Use one of the symbols <, >, or = to compare the two numbers to compare the two numbers of e-mails sent. Explain how you used the digits to determine your answer

Answers

Answer:

Last month (2,014) = (2,014) Month before last month

Your answer would be 2,014 = 2,014

The digits are all the same. Therefore, I concluded that the numbers were the same. This means that a equal sign should be used.

Hope this helps. Pls mark brainliest. :S

what is 1*1000000*5*6/3*1757/4*45567-4-6-8*5846584*0+1+2+3+4+5+6+7+8+9+10

Answers

Answer:

2.0015305e+15 is the answer

Decrease £110 by 50%

Answers

Half of 110 is 55 so the answer is 55

Answer:

£55

Step-by-step explanation:

50% is just another way to say 1/2. This means that 1/2 of £110 is £55.

5x squared +1=4x-7 solve the equation using the quadratic formula

Answers

Answer:

Step-by-step explanation:

      - B  ±  √ B2-4AC

 x =   ____________

                     2A

which will be

          4 ± √ -144

  x  =    _______

                     10

x =(4-√-144)/10=(2-6i)/5= 0.4000-1.2000

Given an undirected graph G= (V, E) determined by the adjacency matrix as follows:
01001

10111

01010

01101

11010

The number of odd degree vertices of the graph is

Select one

A.2
B.4
C.3
D.1
if anyone knows the question please help me :(

Answers

The degree of a vertex is defined as the number of edges that touch the vertex. The (i, j)-th entry of the adjacency matrix tells you whether vertex i touches vertex j (1 if yes, 0 if no). Then the degree of vertex i is equal to the sum of the i-th row in the adjacency matrix.

• vertex 1 : degree = 0 + 1 + 0 + 0 + 1 = 2

• vertex 2 : degree = 1 + 0 + 1 + 1 + 1 = 4

• vertex 3 : degree = 0 + 1 + 0 + 1 + 0 = 2

• vertex 4 : degree = 0 + 1 + 1 + 0 + 1 = 3

• vertex 5 : degree = 1 + 1 + 0 + 1 + 0 = 3

So, the correct answer is A. 2.

Explain this picture ​

Answers

my spine :)))))))))))))


Someone help me pls

Answers

Answer:

248 is your answer

Step-by-step explanation:

The two angles are supplementary, therefore together they add to 360. So take 360 and subtract 112, you get 248. Hope this helps you :)


10. Flora rented a space at the flea market.
Her space included electricity.

Booth Space
each day:$25

Electricity:$2


Which expression gives the total cost of
the day's rental if she used h hours of
electricity?

A. 25-2h
B. 25 + 2h
C.(25+2) x h
D. 25h+2

Answers

Answer:

It is D

Step-by-step explanation:

I am not exactly sure how to explain but i'll try.

The 2 dollars is added to her price, so it is not A.

The 25 dollars is every day, so it is mutlipled by x. So its not B.

The 2 dollars is not everyday like the 25 dollars, so its not C b/c its not everyday.

-2(x-6)=6 help me out

Answers

Answer:

x = 9

Step-by-step explanation:

i took the quiz

[tex]\mathfrak{-2(x-6)=6} [/tex]

[tex]\mathfrak{-2x+12=6} [/tex]

[tex]\mathfrak{-2x=6-12} [/tex]

[tex]\mathfrak{-2x=-6} [/tex]

[tex]\boxed{\mathfrak{x=3} }[/tex]

1/2 2/5 6/10 least to greatest

Answers

Answer:

2/5, 1/2, 6/10

Step-by-step explanation:

did problem

Answer:

1. 2/5

2. 1/2

3. 6/10

Step-by-step explanation:

Hope this helps

May I get braineist pls

Solve


5 + 3 ( x - 1 ) = 5x - 6​

Answers

The answer is x=4

5+3(x-1)=5x-6
Do the parentheses
5+3x-3=5x-6
Combine like terms
3x+2=5x-6
Subtract 3x to both sides
2=2x-6
Add 6 to both sides
8=2x
Divide by 2 on both sides
4=x

3.) Jason has one less dimes than quarters and twice as many nickels

as quarters. If he has a total of $4.40, how many of each type

coin does he have?

Answers

Answer:

10 quarters, 9 dimes, and 20 nickels.

Step-by-step explanation:

q - 1 = d, 2q = n or q = 1/2n

.25q + .10(q - 1) + .05(2q) = 4.40

.25q + .10q - .10 + .10q = 4.40

.45q - .10 = 4.40

.45q = 4.50

4.5/.45 = 10

He has 10 quarters, 9 dimes, and 20 nickels.

The ratio of the number of cups of red paint to the number of cups of yellow paint that Garcia mixed to make a shade of orange paint is 4:1.

Part A: Write a sentence to describe the number of cups of red paint that Garcia mixed with each cup of yellow paint. (4 points)

Part B: How many cups of yellow paint should Garcia add to each cup of red paint to make the shade of orange? Explain your reasoning. (4 points)

Part C: If Garcia purchased 6 cups of red paint for $42, what is the unit price per cup of red paint? (2 points)



Pls answer i need help

Answers

Answer:

Uggghhh

Step-by-step explanation:

I dont know

Answer:

Part A: Garcia mixed 4 cups of red with 1 cup of yellow.

Part B: 1/4 because 4 cups of red requires 1 cup of yellow, so 1 divided by 4 is 1/4.

Part C: $7

Hope this helps! :)

Step-by-step explanation:

A middle school took all of its 6th grade students on a field trip to see a symphony at a theater that has 4500 seats. The students filled 2205 of the seats in the theater. What percentage of the seats in the theater were filled by the 6th graders on the trip?

Answers

Answer:

49%.

Step-by-step explanation:

2205      100        220500

........... x ..........  =  ..................  = 49 (49%).

4500        1              4500

The theater filled up with 49% of seats with 6th grade students.

What is a expression? What is a mathematical equation? What is Equation Modelling?

A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

We have a middle school that took all of its 6th grade students on a field trip to see a symphony at a theater that has 4500 seats. The students filled 2205 of the seats in the theater.

Assume that the theater filled up [x]% of seats at the theater. Then, we can write -

[x]% = (2205/4500) x 100

[x]% = 0.49 x 100

[x]% = 49%

Therefore, the theater filled up with 49% of seats with 6th grade students.

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Can someone help me please?

Answers

Answer:

put your equation into fractions and then see what or which 1mathces with the image

Step-by-step explanation:

PLEASE HELP ME SOLVE THIS!!!​

Answers

Answer:

c. m < 13

Step-by-step explanation:

13 - 7 is equal to 6 so m must be less than 13 to be less than 6.

Find two numbers whose sum 7500 and whose product is the maximum value.
(Round your answer to the nearest whole number.)
Blank = 1
Blank = 2

Answers

The two numbers whose sum is 7500 and whose product  is a maximum value are 3750 and 3750.

Let the two numbers be x and y.

Since their sum equals 7500,

x + y = 7500  (1)

Their product f(x,y) = xy  (2)

We desire to maximize the product.

From (1), x = 7500 - y

Substituting x into (2), we have

f(x,y) = xy

f(x,y) = (7500 - y)y

f(y) = 7500y - y²

To maximize the product, we find the value of y that maximizes f(y), we differentiate f(y) and equate it to zero.

So, df(y)/dy = d(7500y - y²)/dy

f'(y)/dy = d(7500y)/dy - dy²/dy

f'(y)/dy = 7500 - 2y

Equating it to zero, we have

7500 - 2y = 0

7500 = 2y

y = 7500/2

y = 3750

To determine if this gives a maximum for f(y), we differentiate f'(y) with respect to y.

So, f"(y) = d(7500 - 2y)dy

f"(y) = 0 - 2

f"(y) = -2 < 0.

So, y = 3750 gives a maximum for f(y) = f(x, y)

Since x = 7500 - y

Substituing y = 3750 into the equation, we have

x = 7500 - 3750

x = 3750

So, the two numbers whose sum is 7500 and whose product is a maximum product are 3750 and 3750.

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i have 100 points
i need some help!!

Answers

Answer:

Option D:  [tex]\displaystyle\mathsf{-\frac{4}{5}\:\leq\:x}[/tex]

Step-by-step explanation:

GIven the following inequality statement:

[tex]\displaystyle\mathsf{7x\:+\:6\:\geq \:\frac{-x}{2}}[/tex]

Start by eliminating the fraction on the right-hand side.

Multiply both sides of the inequality statement by 2:

[tex]\displaystyle\mathsf{2(7x\:+\:6)\:\geq \:\bigg[\frac{-x}{2}\bigg](2)}[/tex]

2(7x + 6) ≥ -x

Next, distribute 2 into the terms inside the parenthesis:

14x + 12 ≥ -x

Subtract 12 from both sides:

14x + 12 - 12 ≥ -x - 12

14x  ≥ -x - 12

Add x to both sides:

14x + x  ≥ -x + x - 12

15x ≥ - 12

Divide both sides by 15 to isolate x:

[tex]\displaystyle\mathsf{\frac{15x}{15}\:=\:\frac{-12}{15}}[/tex]

[tex]\displaystyle\mathsf{x\geq -\frac{4}{5}}[/tex] or  [tex]\displaystyle\mathsf{-\frac{4}{5}\:\leq\:x}[/tex]

Therefore, the correct answer is Option D:  [tex]\displaystyle\mathsf{-\frac{4}{5}\:\leq\:x}[/tex]

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