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
He school that Tony goes to is selling tickets to a play. On the first day of ticket sales the school
UNO [17]

Answer:

The price of 1 adult ticket is 12 dollars, and the price of a ticket for one student is 7 dollars

Step-by-step explanation:

Make a system of equations for the two days that the play was shown.

Let x = the price of an adult ticket

Let y = the price of a student ticket

For the first day:

<h3>9x+8y=164</h3>

For the second day:

<h3>2x+7y=73</h3>

Now, we can solve using the elimination method. Multiply the first equation by 2 and the second equation by 9. Then swap the order of the equations.

<h3>18x+63y= 657</h3><h3>-</h3><h3>18x+16y= 328</h3><h3>0x+ 47y= 329</h3><h3>divide both sides by 47</h3><h3>y = 7</h3><h3>Plug in 7 for y for the 2nd equation</h3><h3>2x+7(7)=73</h3><h3>2x+49=73</h3><h3>subtract 49 from both sides</h3><h3>2x= 24</h3><h3>divide both sides by 2</h3><h3>x = 12 </h3><h3>Check:</h3><h3>2(12)+7(7)=73</h3><h3>24+49= 73!</h3>
8 0
2 years ago
Read 2 more answers
A student solved this problem by working backward. Jared has some flowers. He has 6 more red flowers than yellow flowers. He has
goldenfox [79]
5 + 7 = 12 silver balloons
12 x 3 = 36 bronze
12 + 36 + 5 = 53
4 0
3 years ago
A die is rolled one thousand times. The percentage of aces should be around ____ or so. the first step iin computing this proble
Katyanochek1 [597]

Answer:

Following are the solution to the given question:

Step-by-step explanation:

Please find the complete question in the attached file.

Box \ average =\frac{1}{6}  \\\\SD = \frac{1}{6}  \times \frac{5}{6} = \frac{5}{36} \approx 0.373\\\\Expected \ value= 1000 \times \frac{1}{6}  \approx 166.7\\\\SE = 1000 \times 0.373 \approx 11.8\\\\

The total number of aces is approximately 167, 12 or more. The SE is 12 out of 1000, which is 1.2 percent.

The percentage of aces is expected to be around 16.67% , or 1.2%.

7 0
3 years ago
Bro no balls youll give me 1$ on Paypal ill give you a whole 100 points :}
IgorC [24]

Answer:

hmmmmmmm

Step-by-step explanation:

nah I am good thx for the free points

7 0
2 years ago
Read 2 more answers
What’s the equation.
cestrela7 [59]

Answer:

c=42

s=30

Step-by-step explanation:

5 0
2 years ago
Other questions:
  • Jonathan's car used 3 gallons to travel 45 miles. How many miles can the car go for each gallon of gas?
    11·2 answers
  • Gordon wants to buy 3 2/5 cups of almonds. There are 1 7/10 cups of almonds in each package. How many packages of almonds should
    9·1 answer
  • Sarah cut off 1/8 of a 48-inch piece of ribbon. Jane cut off 1/9 of a 36-inch piece of ribbon. Whose piece is longer? How much l
    13·1 answer
  • What is the area of the blue glass will be needed to fill each butterfly dot if the distance around each dot it 24pie inches
    6·2 answers
  • Which of these best describes the Pythagorean theorem?
    15·1 answer
  • Which statement is true???
    9·2 answers
  • A soccer team has two sports coolers that hold 5 gallons each coach Fenimore has 12 fluid oz cups for the team to drink from. Ho
    14·2 answers
  • SOMEONE HELP I'LL MARK CORRECT ANSWER BRAINLY
    11·1 answer
  • One fourth of a cake was shared equally among 3 people.
    9·2 answers
  • Someone please help me with this question And you can get 20 points
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!