1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
solniwko [45]
3 years ago
6

If a/b < c/d with b > 0, d > 0, prove that a+c / b+d lies between the two fractions a/b and c/d .​

Mathematics
1 answer:
Tems11 [23]3 years ago
5 0

Divide through everything by <em>b</em> :

\dfrac{a+c}{b+d} = \dfrac{\dfrac ab + \dfrac cb}{1 + \dfrac db}

Since <em>a/b</em> < <em>c/d</em>, it follows that

\dfrac{a+c}{b+d} < \dfrac{\dfrac cd+\dfrac cb}{1 + \dfrac db}

Multiply through everything on the right side by <em>b/d</em> to get

\dfrac{a+c}{b+d} < \dfrac{\dfrac{bc}{d^2}+\dfrac cd}{\dfrac bd+1} = \dfrac{\dfrac cd\left(\dfrac bd+1\right)}{\dfrac bd+1} = \dfrac cd

and so (<em>a</em> + <em>c</em>)/(<em>b</em> + <em>d</em>) < <em>c/d</em>.

For the other side, you can do something similar and divide through everything by <em>d</em> :

\dfrac{a+c}{b+d} = \dfrac{\dfrac ad + \dfrac cd}{\dfrac bd + 1}

and <em>a/b</em> < <em>c/d</em> tells us that

\dfrac{a+c}{b+d} > \dfrac{\dfrac ad + \dfrac ab}{\dfrac bd + 1}

Then

\dfrac{a+c}{b+d} > \dfrac{\dfrac ab + \dfrac{ad}{b^2}}{1 + \dfrac db} = \dfrac{\dfrac ab\left(1+\dfrac db\right)}{1 + \dfrac db} = \dfrac ab

and so (<em>a</em> + <em>c</em>)/(<em>b</em> + <em>d</em>) > <em>a/b</em>.

Then together we get the desired inequality.

You might be interested in
At a certain time of day, a tree that is x meters tall casts a shadow that is x−46x−46 meters long. if the distance from the top
Alexus [3.1K]
By the Pythagorean theorem,
.. x^2 +(x -46)^2 = (x +4)^2
.. 2x^2 -92x +2116 = x^2 +8x +16
.. x^2 -100x +2100 = 0
.. (x -50)^2 -400 = 0
.. x = 50 ±20 . . . . . only x=70 makes any sense in the problem

The height, x, of the tree is 70 meters.
5 0
3 years ago
What percent of 88 is 72.6? ...?
Feliz [49]
<span>Percentage problems are simple division problems. Here, what percentage of x is y means the answer is y divided by x (times 100). Let's apply it to this question.
Divide 72.6 by 88, 72.6/88. So the answer would be 0.825 then multiply it to 100 to make it a whole. The answer would 82.5%. I hope this helps you with your problem</span>
8 0
3 years ago
Will mark you as brainliest. If your good with polynomials.
kobusy [5.1K]
To solve/simplify this all you have to do is group like terms (the x^2's with each other, the x's with each other, and the normal numbers, -8)

14x^2-8+5x-6x^2+2x

group the x^2 (add 14x^2 to -6x^2)

8x^2-8+5x+2x

group the x's together (add 5x and 2x together)

8x^2+7x-8

Your answer will be d) 8x^2+7x-8



4 0
3 years ago
A car's velocity changes at a rate of 36 miles per hour in 0.5 hour.
kotegsom [21]

Answer:

72

Step-by-step explanation:

In order to find the acceleration we need first to defined it, so:

acceleration=((final velocity)-(initial velocity))/time interval

a=(vf-vi)/dt

But (vf-vi) actually represents the velocity change, so (vf-vi)/dt represents the velocity change rate. This means that in our case:

a=(36miles/hour)/(0.5hours)

a=72miles/hour^2

In conclusion the acceleration is a=72miles/hour^2, without units just 72.

7 0
3 years ago
Help with practice problem
padilas [110]

Answer:

attached above with detail explanation.

5 0
2 years ago
Other questions:
  • What is the explicit formula for this geometric sequence?<br><br>8,4,2,1
    13·1 answer
  • Explain the different ways it is possible to add two rational numbers and get a negative number
    7·2 answers
  • What is the slope and y-intercept of the graphed line?
    8·2 answers
  • Express each ratio as a fraction in simplest form. Hint: 1 yard = 3 feet.<br><br> 9 yards to 48 feet
    13·1 answer
  • Twice the sum of six "x" and seventy eight.
    8·1 answer
  • What is the RANGE of the graph?
    6·1 answer
  • 2/20 : 1/40 as a unit rate
    10·1 answer
  • Due to gravity, an object falls 16t² feet in t seconds. You drop a backpack from a window that is 100 feet above the ground. Wil
    8·1 answer
  • Find the area of the shaded region.
    7·1 answer
  • Please help me this is so annoying and I need it done really fast so if you could just put an quick answer no explanation just t
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!