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Tutors Answer Your Questions about Evaluation Word Problems (FREE)
Question 1108035: Donna wants to make trail mix made up of almonds, walnuts, and raisins. She wants to mix one part almonds, two parts walnuts, and three parts raisins. Almonds cost $12 per pound, walnuts cost $9 per pound, and raisins cost $5 per pound.
Donna has $15 to spend on the trail mix. Determine how many pounds of trail mix she can make. ( have an algebraic solution)
Found 3 solutions by n2, josgarithmetic, ikleyn:
Answer by n2(79)  (Show Source):
You can put this solution on YOUR website! .
Donna wants to make trail mix made up of almonds, walnuts, and raisins. She wants to mix one part almonds,
two parts walnuts, and three parts raisins. Almonds cost $12 per pound, walnuts cost $9 per pound,
and raisins cost $5 per pound.
Donna has $15 to spend on the trail mix. Determine how many pounds of trail mix she can make. (have an algebraic solution)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let's combine 1 pound of almonds, 2 pounds of walnuts, and 3 pounds of raisins.
This combination is the prescribed proportion. The total mass of this mixture is 1 + 2 + 3 = 6 pounds,
and its total cost is 1*$12 + 2*$9 + 3*$5 = $45.
So, 6 pounds of this mixture cost 45 dollars.
$15 dollars that Donna has is 1/3 of $45 - - - hence,
for $15 Donna may have 1/3 of 6 pounds, i.e. 2 pounds of the mixture.
ANSWER. For $15, Donna may have 2 pounds of the mix.
Answer by josgarithmetic(39792) (Show Source):
Answer by ikleyn(53748)  (Show Source):
You can put this solution on YOUR website! .
Donna wants to make trail mix made up of almonds, walnuts, and raisins. She wants to mix one part almonds,
two parts walnuts, and three parts raisins. Almonds cost $12 per pound, walnuts cost $9 per pound,
and raisins cost $5 per pound.
Donna has $15 to spend on the trail mix. Determine how many pounds of trail mix she can make. (have an algebraic solution)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Setup equation in the post by @mananth is incorrect, leading to incorrect answer of 4 pounds of trail mix.
So, his solution is incorrect both technically and conceptually.
In my post below, I will give simple and short straightforward solution.
Let's combine 1 pound of almonds, 2 pounds of walnuts, and 3 pounds of raisins.
This combination is the prescribed proportion. The total mass of this mixture is 1 + 2 + 3 = 6 pounds,
and its total cost is 1*$12 + 2*$9 + 3*$5 = $45.
So, 6 pounds of this mixture cost 45 dollars.
$15 dollars that Donna has is 1/3 of $45 - - - hence,
for $15 Donna may have 1/3 of 6 pounds, i.e. 2 pounds of the mixture.
ANSWER. For $15, Donna may have 2 pounds of the mix.
Solved correctly.
Question 353429: The sumof the squares of three consecutive positive integers is 77. Find the integers.
The book gives a hint: if one integer is x, the next consecutive positive integer is x+1, and the third is x+2
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
The sum of the squares of three consecutive positive integers is 77. Find the integers.
The book gives a hint: if one integer is x, the next consecutive positive integer is x+1, and the third is x+2
~~~~~~~~~~~~~~~~~~~~~~~~~
There is much more elegant way to solve, than the way assumed in the book.
Let 'n' be the central number of these three consecutive integers.
So, our numbers are (n-1), n and (n+1).
Make an equation for the sum of squares
(n-1)^1 + n^2 + (n+1)^2 = 77,
(n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1) = 77,
Cancel opposite terms '-2n' and '2n' and combine like terms
3n^2 + 2 = 77 ---> 3n^2 = 77 - 2 ---> 3n^2 = 75 ---> n^2 = 75/3 = 25 ---> n =  = +/- 5.
Since we are looking for positive integer numbers, they are 4, 5 and 6. ANSWER
Solved, without necessity to solve a quadratic equation.
Question 420867: If I have a 20 qt radiator containing a 80% antifreeze solution. How much of the solution should I drain and replace with pure water to get a solution that is 50% antifreeze?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792) (Show Source):
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
If I have a 20 qt radiator containing a 80% antifreeze solution, how much of the solution
should I drain and replace with pure water to get a solution that is 50% antifreeze?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The solution in the post by @mananth is incorrect.
I came to bring a correct solution.
Let x be the volume of the original 80% antifreeze solution to partly drain
and to replace with pure water to get the 40% antifreeze solution.
After draining, we then have (20-x) quartz of the 80% antifreeze solution.
It contains 0.8*(20-x) quartz of the pure antifreeze.
Adding water does not change the amount of the antifreeze in the solution.
At the end, the volume of the pure antifreeze in the radiator after adding x quartz of water is 0.5*20 quartz.
So, we equate these two expressions for the pure antifreeze amount
0.8*(20-x) = 0.5*20 quartz. (1)
Simplify and find x
16 - 0.8x = 10,
16 - 10 = 0.8x,
6 = 0.8x
x =  =  = 7.5.
ANSWER. 7.5 quartz of the original 80% solution should be drained and replaced by pure water to get the 50% antifreeze solution.
Solved correctly.
Question 476566: Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test?)
Found 6 solutions by AnlytcPhil, Edwin McCravy, mccravyedwin, greenestamps, josgarithmetic, ikleyn:
Answer by AnlytcPhil(1810)  (Show Source):
You can put this solution on YOUR website!
josgarithmetic(39630) misread the 4 as a 3
Here is a correction of his solution, for his benefit.
let x = number of questions that are 2 points each.
let y = number of questions that are 4 points each.
since the total number of questions is 35, then your first equation is:
x + y = 35
since each x gets you 2 points and each y gets you 4 points, then your second equation is:
2x + 4y = 100
solve these 2 equations simultaneously for your answer.
use second equation to solve for y in terms of x.
you get y = 35 - x
substitute for y in the first equation to get:
2x + 4(35-x) = 100 which becomes:
2x + 140 - 4x = 100 which becomes:
-2x + 140 = 100
add 2x to both sides of this equation....
140 = 100 +2x
and subtract 100 from both sides of this equation to get:
40 = 2x
20 = x
since x + y = 35, this means that y = 15
you have 20 questions that are 2 points each and 15 questions that are 4 points each for a total of 20 + 15 = 35 questions with a total of 40 + 60 = 100 points.
Edwin
Answer by Edwin McCravy(20077)  (Show Source):
You can put this solution on YOUR website!
You might also enjoy comparing this other solution to that problem:
Mrs. Jones has 35 animals, all sheep and ducks, on her farm. The animals
have a total of 100 legs and 35 heads. How many sheep and how many ducks
are on her farm?
(1) If all 35 animals were sheep, the total number of legs would be
35*4 = 140, which is 40 legs more than the actual total of 100 legs.
So we need to reduce the number of legs by 40.
(2) Each duck has 2 legs .
(3) To reduce the number of legs by an additional 40 legs, the number of
ducks needed is 40/2 = 20 ducks.
(4) So there are 20 ducks and 35-20 = 15 sheep.
ANSWER: 15 sheep and 20 ducks.
CHECK: 15(4) + 20(2) = 60+40 = 100
Edwin
Answer by mccravyedwin(421)  (Show Source):
You can put this solution on YOUR website!
I thought you might enjoy comparing Greenestamps' solution to this problem:
Mrs. Jones has 35 animals, all sheep and ducks, on her farm. The animals
have a total of 100 legs and 35 heads. How many sheep and how many ducks
are on her farm?
(1) If all 35 animals were ducks, the total number of legs would be only
35*2 = 70, which is 30 legs less than the actual total of 100 legs.
So we need another 30 legs.
(2) Each sheep has 2 more legs than each duck.
(3) To get the additional 30 legs, the number of sheep needed is 30/2 = 15.
(4) So there are 15 sheep and 35-15 = 20 ducks.
ANSWER: 15 sheep and 20 ducks.
CHECK: 15(4) + 20(2) = 60+40 = 100
Edwin
Answer by greenestamps(13327)  (Show Source):
You can put this solution on YOUR website!
Here is a way to solve the problem mentally, using logical reasoning instead of formal algebra. The calculations performed are nearly identical to those used in the formal algebraic solution shown by the other tutor.
(1) If all 35 questions were worth 2 points each, the total number of points would be 35*2 = 70, which is 30 less than the actual total of 100. So we need another 30 points.
(2) Each 4-point question is worth 2 points more than each 2-point question.
(3) To get the additional 30 points, the number of 4-point questions needed is 30/2 = 15.
(4) So there are 15 4-point questions and 35-15 = 20 2-point questions.
ANSWER: 15 4-point questions and 20 2-point questions.
CHECK: 15(4) + 20(2) = 60+40 = 100
Answer by josgarithmetic(39792) (Show Source):
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems.
Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test?)
~~~~~~~~~~~~~~~~~~~~~~~~~~~
The solution in the post by @Theo is incorrect.
His equations are not adequate to the problem, and the answer is wrong.
I came to bring a correct solution.
Let x be the number of the 2-points problems and
let y be the number of the 4-points problems.
Write equations as you read the problem
x + y = 35, (1)
2x + 4y = 100. (2)
It can be solved by different methods: by the Substitution method or by the Elimination method.
I will solve it here using the Elimination method.
Multiply equation (1) by 2 (both sides). Keep equation (2) as is.
You will get
2x + 2y = 70, (1')
2x + 4y = 100. (2')
Now subtract equation (') from equation (2').
The terms with '2x' will annihilate each other, and you will get
4y - 2y = 100 - 70,
2y = 30,
y = 30/2 = 15.
So, y = 15. To find x, substitute this value y = 15 in equation (1).
You will get
x + 15 = 35 ---> x = 35 - 15 = 20.
ANSWER. There are 20 2-points problems and 15 4-points problems.
CHECK. Checking is a necessary part of the solution.
Substitute the found values x and y into equation (1) to make sure
that the left side value coincides with the right side value.
Substitute the found values x and y into equation (2) to make sure
that the left side value coincides with the right side value.
At this point, the solution is completed in full.
/////////////////////////////////////////
There is no any need to recheck the @josgarithmetic' solution,
since he simply re-wrote it from my solution, with variations.
Question 1185401: Shreya is very interested in cryogenics (the science of very low temperatures). With the help of her science teacher she is doing an experiment on the effect of low temperatures on bacteria. She cools one sample of bacteria to a temperature of -51°C, another to -45oC, a third to -81oC, and finally another to -76°C. What was the temperature difference in the two coldest experiments? Each experiment began with 12 individual bacteria, and the strain of cold resistant bacteria she was testing stopped growing at -47oC, and they doubled each minute. Further, the bacteria started to die off at 75oC, losing ⅓ of the colony each minute. How many bacteria will Shreya have at the end of the third experiment if she began the experiment at 13oC, and it takes 15 seconds to increase or decrease the temperature by 1 degree?
Answer by CPhill(2189)  (Show Source):
You can put this solution on YOUR website! Here's how to break down this problem:
1. **Temperature Difference:** The two coldest temperatures are -81°C and -76°C. The difference between them is |-81 - (-76)| = |-81 + 76| = |-5| = 5°C.
2. **Bacteria Growth (Irrelevant to the question asked):** The information about the bacteria doubling and dying off, and the temperature at which they stop growing or start dying is extra information and not needed to answer the temperature difference question.
3. **Bacteria in the third experiment:**
* **Initial Temperature:** 13°C
* **Target Temperature:** -81°C
* **Temperature Difference:** 13 - (-81) = 94°C
* **Time to cool:** 94°C * 15 seconds/°C = 1410 seconds
* **Minutes to cool:** 1410 seconds / 60 seconds/minute = 23.5 minutes
Since the bacteria stop growing at -47°C, we need to determine how long it takes to reach that temperature:
* Temperature difference to reach -47°C: 13 - (-47) = 60°C
* Time to reach -47°C: 60°C * 15 seconds/°C = 900 seconds = 15 minutes.
For the first 15 minutes, the bacteria doubles every minute. So, after 15 minutes, the number of bacteria will be 12 * 2^15 = 393216.
The bacteria stops growing at -47°C. The experiment cools to -81°C, which is much lower, so the bacteria will not grow beyond that point. The time to reach -81°C is 23.5 minutes, but the bacteria count will be the same as after 15 minutes because the bacteria stop growing at -47°C.
Therefore, Shreya will have 393,216 bacteria at the end of the third experiment.
Question 1196669: DATA LINK: https://imagizer.imageshack.com/img924/7537/If85YO.png
https://imagizer.imageshack.com/img922/9241/hUOEdV.png
Is economic growth in the developing world related to growth in the industrialized countries? A scatterplot of the growth (in % of Gross Domestic Product) of the least developed countries (as classified by the UN) vs. the growth of a number of industrialized countries is provided. Each point represents a year from the past few decades. The output of a regression analysis is also provided. Complete parts a through c below.
Answer by ElectricPavlov(122) (Show Source):
You can put this solution on YOUR website! **a) Describe the relationship between economic growth in the least developed countries and the growth in industrialized countries.**
* **Examine the Scatterplot:**
* **Positive Relationship:** If the scatterplot shows a general upward trend (points sloping upwards from left to right), it suggests that economic growth in industrialized countries tends to be associated with economic growth in the least developed countries.
* **Negative Relationship:** If the scatterplot shows a downward trend, it indicates that growth in industrialized countries is associated with slower growth in the least developed countries.
* **No Relationship:** If the points are scattered randomly with no clear trend, it suggests little or no relationship between the two.
* **Consider the Regression Analysis Output:**
* **Coefficient of the Industrialized Country Growth:**
* A positive coefficient suggests a positive relationship.
* A negative coefficient suggests a negative relationship.
* **Significance of the Coefficient:**
* A statistically significant coefficient (usually indicated by a low p-value) provides evidence that the relationship between the two growth rates is not due to random chance.
**b) Interpret the slope of the regression line.**
* **Slope:** The slope of the regression line represents the estimated change in the growth rate of the least developed countries for each 1% increase in the growth rate of industrialized countries.
* **Example:** If the slope is 0.5, it suggests that for every 1% increase in the growth rate of industrialized countries, the growth rate of the least developed countries is estimated to increase by 0.5%.
**c) Discuss the limitations of using this analysis to predict the growth of the least developed countries.**
* **Correlation does not imply causation:** Even if a strong relationship is found, it doesn't necessarily mean that growth in industrialized countries directly causes growth in the least developed countries. Other factors could be influencing both.
* **External factors:** Global economic shocks, political instability, natural disasters, and other external factors can significantly impact growth in both developed and developing countries, which may not be fully captured by the model.
* **Heterogeneity among countries:** The analysis treats all least developed countries as a single group, ignoring the significant diversity within this group in terms of economic structure, resources, and development levels.
* **Time-varying relationships:** The relationship between the growth rates of developed and developing countries may not be constant over time. Factors like globalization, technological advancements, and changing trade patterns can alter the nature of this relationship.
**In summary:**
* The scatterplot and regression analysis can provide insights into the potential relationship between economic growth in industrialized and least developed countries.
* However, it's crucial to interpret the results cautiously and consider the limitations of the analysis.
**Note:** To provide more specific answers, I would need to see the actual scatterplot and the regression analysis output.
Question 1205323: Siti has RM450000 in her ASB. She wants to invest in Gading Mutual deposit, Maju
Makmur bar gold, Indah certificate deposit and Selamat Maju bar gold which pay
simple annual interest of 9%, 6%, 10% and 15%, respectively. Moreover, she wants
to combine annual return of 8% and want to have only one-third of investment in
Indah certificate deposit and Selamat Maju bar gold.
a. Write the linear model system equation for the whole investment.
c. Use elimination method to find each of the investment
Answer by CPhill(2189) (Show Source):
You can put this solution on YOUR website! ## Setting up the Linear Model
**Let's denote:**
* **G:** Amount invested in Gading Mutual Deposit
* **M:** Amount invested in Maju Makmur Bar Gold
* **I:** Amount invested in Indah Certificate Deposit
* **S:** Amount invested in Selamat Maju Bar Gold
**Based on the given information, we can set up the following system of linear equations:**
1. **Total Investment:** G + M + I + S = 450000
2. **Total Annual Return:** 0.09G + 0.06M + 0.10I + 0.15S = 0.08 * 450000
3. **Indah and Selamat Maju Investment:** I + S = (1/3) * 450000
## Solving the System Using Elimination Method
**Step 1: Simplify the Equations**
* Equation 2: 9G + 6M + 10I + 15S = 360000
* Equation 3: I + S = 150000
**Step 2: Eliminate a Variable**
* Let's eliminate S. From Equation 3, we can express S as:
* S = 150000 - I
* Substitute S in Equations 1 and 2:
* Equation 1: G + M + I + (150000 - I) = 450000
=> G + M + 150000 = 450000
* Equation 2: 9G + 6M + 10I + 15(150000 - I) = 360000
=> 9G + 6M - 5I = -750000
**Step 3: Simplify the Equations Further**
* Equation 1: G + M = 300000
* Equation 2: 9G + 6M - 5I = -750000
**Step 4: Eliminate Another Variable**
* Let's eliminate M. Multiply Equation 1 by -6 and add it to Equation 2:
* -6G - 6M = -1800000
* 9G + 6M - 5I = -750000
* -------------------------
* 3G - 5I = -2550000
**Step 5: Solve for G and I**
* From Equation 1, we can express M as:
* M = 300000 - G
* Substitute M in the simplified Equation 2:
* 3G - 5I = -2550000
* 3G - 5(150000 - G) = -2550000
* 8G = 1200000
* G = 150000
* Substitute G in M = 300000 - G:
* M = 300000 - 150000 = 150000
* Substitute G in I + S = 150000:
* I + S = 150000
* I + (150000 - I) = 150000
* S = 0
Therefore, the investments are:
* Gading Mutual Deposit: RM150,000
* Maju Makmur Bar Gold: RM150,000
* Indah Certificate Deposit: RM150,000
* Selamat Maju Bar Gold: RM0
Question 1209141: Carly just opened her own nail salon. Based on experience, she knows that her daily profit, P, in dollars, can be modelled by the relation P=-15x^2+240x-640, where x is number of clients per day. How many clients should she book each day to maximize her profit?
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
Carly just opened her own nail salon. Based on experience, she knows that her daily profit, P,
in dollars, can be modelled by the relation P=-15x^2+240x-640, where x is number of clients
per day. How many clients should she book each day to maximize her profit?
~~~~~~~~~~~~~~~~~~~~
The given function is a quadratic function.
Since the coefficient at x^2 is negative, the parabola is downward and has a maximum.
The maximum value is achieved a  =  , where "a" is the coefficient at x^2
and "b" is the coefficient at x. In your case,
 =  =  = 8.
Hence, the profit is maximum if Carly books 8 clients per day. ANSWER
Solved.
--------------------
On finding the maximum/minimum of a quadratic function see the lessons
- HOW TO complete the square to find the minimum/maximum of a quadratic function
- Briefly on finding the minimum/maximum of a quadratic function
- HOW TO complete the square to find the vertex of a parabola
- Briefly on finding the vertex of a parabola
Consider these lessons as your textbook, handbook, tutorials and (free of charge) home teacher.
Learn the subject from there once and for all.
Question 1208966: the numbers on the houses on the north side of carlito street are consecutive odd starting with 1. The plastic digits used to label each house are $0.04 per digit. If it costs $58.12 to label all the houses on the north side of Carlito street, how many houses are there?
Found 3 solutions by Alan3354, mccravyedwin, ikleyn:
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! Not a solution the this problem, but:
-------------
Our address at work is 910 Industrial Blvd.
A guy game in, asked for help finding his customer.
Asked me where 909 Industrial was.
Obviously, it was directly across the street.
I told him that, went out front and pointed to it.
He was grateful.
He was driving a "big rig," 18 wheeler.
===============
How can a person making deliveries for living not know about odd and even addresses?
I considered explaining that to him, but decided not to.
Probably then, "How do you know if an address is odd or even?"
There is no lower limit to intelligence.
Answer by mccravyedwin(421) (Show Source):
You can put this solution on YOUR website!
Let's pretend they put plastic digits on ALL the houses on both sides of the
street. and that there are the same number of houses on both sides of the
street.
The number of digits in all the even integers 0,2,...,2n is the same as the
number of digits in all the odd integers 1,3,...,2n+1
It would have cost twice as much, $58.12x2 = $116.24, to put plastic digits on
all the houses on both the north and south sides of the street, that is, IF the
first even-numbered house were numbered 0.
[Even though 0 is not normally used as a house number, let's pretend it is
on this street anyway.]
If we consider 0 as a 1-digit number, then
there are 10 1-digit numbers which would cost $0.04x10=$0.40,
there are 100-10=90 2-digit numbers which would cost $0.04x2x90=$7.20,
and there are 1000-100=900 3-digit numbers which would cost $0.04x3x900=$108.
That would cost $0.40+$7.20+$108=$115.60
Since $116.24-115.60=$0.64, that means they placed $0.64/$0.04 = 16 more
plastic digits than they placed on houses numbered with 3 or fewer digits.
So 16 plastic digits were placed on 4 houses with 4-digit numbers.
These are the houses numbered 1000, 1001, 1002, and 1003.
So the even house numbers are 0,2,4,...,1002 and there are 1002/2=501 of them.
The odd house numbers are 1,3,5,...,1003 and there are also 501 of them.
Answer: There are 501 odd numbered houses on the north side of the street.
Edwin
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
The numbers on the houses on the north side of Carlito street are consecutive odd starting with 1.
The plastic digits used to label each house are $0.04 per digit. If it costs $58.12 to label
all the houses on the north side of Carlito street, how many houses are there?
~~~~~~~~~~~~~~~~~~~
The number of the digits bought is  = 1453.
Of them, 5*1 = 5 digits are for five 1-digit odd numbers from 1 to 9; 1453-5 = 1448 digits left.
45*2 = 90 digits are for 45 2-digit odd numbers from 11 to 99; 1448-90 = 1358 digits left.
450*3 = 1350 digits are for 450 3-digit odd numbers from 101 to 999; 1358-1350 = 8 digits left.
These 8 digits are for two the 4-digit odd numbers 1001 and 1003 on the north side of Carlito street.
The total number of houses on the north side of Carlito street is 5 + 45 + 450 + 2 = 502. ANSWER
Solved.
////////////////////
Below is my comment regarding Edwin' solution to this problem.
The solution by Edwin, giving the answer 501 for the number of houses
with odd plates on them - is INCORRECT.
The Edwin's error is in the last two sentences in his post before the word Answer :
he incorrectly counted even numbers from 0 to 1002, inclusively,
and incorrectly counted odd numbers from 1 to 1003, inclusively.
The correct count for even numbers/plates from 0 to 1002, inclusively, is 502,
as well as the correct count for odd numbers/plates from 1 to 1003, inclusively, is 502.
Question 1207297: Xavier works in an amusement park and is helping decorate it with strands of lights. This morning, he used a total of 28 strands of lights to decorate 4 bushes and 1 tree. This afternoon, he strung lights on 3 bushes and 1 tree, using a total of 24 strands. Assuming that all bushes are decorated one way and all trees are decorated another, how many strands did Xavier use on each
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
Xavier works in an amusement park and is helping decorate it with strands of lights.
This morning, he used a total of 28 strands of lights to decorate 4 bushes and 1 tree.
This afternoon, he strung lights on 3 bushes and 1 tree, using a total of 24 strands.
Assuming that all bushes are decorated one way and all trees are decorated another,
how many strands did Xavier use on each
~~~~~~~~~~~~~~~~
This problem ideally suits to solve it MENTALLY.
As you read the problem, you should notice in your mind, that the difference
between work done in the morning and in the afternoon is 4-3 = 1 bush and 28-24=4 strands of lights.
So, you conclude that Xavier uses 4 strands of lights to decorate each bush.
Thus this morning he used 4*4 = 16 strands of lights to decorate 4 bushes,
and the rest 28-16 = 12 strands of lights to decorate one tree.
ANSWER. 4 strands of lights for each bush and 12 strands of lights for each tree.
CHECK the number of strands of lights used in the afternoon: 4*3+12 = 24. ! correct !
Solved.
Question 1207277: DR stein bought 30 notebooks 60 pencils and 300 erasers to make identical packages with some notebooks some pencils and some erasers for his students he used everything bought and every student got a package. what is the greatest number of students in Dr stein can have in his class?
Answer by MathLover1(20855)  (Show Source):
Question 1206208: Given the following equality:
g(x) + g(x + 3) = 2x + 5
And:
g(2) + g(8) = 12
Find:
g(5)
Thanks
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
Given the following equality:
g(x) + g(x + 3) = 2x + 5
And:
g(2) + g(8) = 12
Find g(5).
~~~~~~~~~~~~~~~~~~~~~
Take x= 2. Then from the formula g(x) + g(x + 3) = 2x + 5 you have
g(2) + g(5) = 9. (1)
Take x= 5. Then from the formula g(x) + g(x + 3) = 2x + 5 you have
g(5) + g(8) = 15. (2)
Add equations (1) and (2). You will get
g(2) + 2*g(5) + g(8) = 24.
Replace here g(2) + g(8) by 12 (since it is given in the problem). You will get
12 + 2*g(5) = 24 ---> 2*g(5) = 24-12 = 12 ---> g(5) = 12/2 = 6.
ANSWER. g(5) = 6.
Solved.
Question 1206112: In a mixture of copper, tin, and lead, one half of the whole, minus 16 lb, was copper; one third of the whole, minus 12 lb, was tin, and one fourth of the whole, plus 4 lb, was lead: how much did the whole mixture weigh?
Answer by josgarithmetic(39792) (Show Source):
You can put this solution on YOUR website! One can transcribe the description exactly as written,
here using w for the whole quantity of mixture.
Copper (w/2)-16
Tin (w/3)-12
Lead (w/4)+4
WHOLE w
.
.
Another way, the whole mixture is some 12 equal parts, so 12M is this mixture quantity.
Copper 6M-16
Tin 4M-12
Lead 3M+4
WHOLE 12M
Remember M here is a factor.
Equation to solve can be  .
.
.
Question 1205729: A cruise ship had 3 times as many adult passengers as child passengers when it left its first port. At a new port, 116 adults and 50 children boarded the ship. There are now 1,216 more adults than children on the ship. How many children were on the ship at first?
Found 3 solutions by josgarithmetic, greenestamps, MathLover1:
Answer by josgarithmetic(39792) (Show Source):
Answer by greenestamps(13327) (Show Source):
You can put this solution on YOUR website!
Let x = number of children at first
Then 3x = number of adults at first
x+50 = number of children after the new port
3x+116 = number of adults after the new port
The number of adults is now 1216 more than the number of children:
Solve using basic algebra; I leave the work to you. (Re-post, showing your work, if you don't get the answer below.)
ANSWER: 575 children and 1725 adults at first
Answer by MathLover1(20855) (Show Source):
Question 1205480: if a + b = -3 and b - c = 6, find the value of 
Found 3 solutions by ikleyn, math_tutor2020, greenestamps:
Answer by ikleyn(53748) (Show Source):
Answer by math_tutor2020(3835) (Show Source):
You can put this solution on YOUR website!
This is a system of 2 equations with 3 unknowns.
The fact we have more unknowns than equations leads to "infinitely many solutions" for this system.
It turns out that each solution is of the form (a,b,c) = (-9-c,6+c,c) which I explain in a later section below.
Let's say c = 0
b-c = 6
b-0 = 6
b = 6
Then,
a+b = -3
a+6 = -3
a = -3-6
a = -9
Or you could say
(a,b,c) = (-9-c,6+c,c)
(a,b,c) = (-9-0,6+0,0)
(a,b,c) = (-9,6,0)
Therefore,
------------------------------------------------------------
Another example
Let c = 1
b-c = 6
b-1 = 6
b = 6+1
b = 7
Then,
a+b = -3
a+7 = -3
a = -3-7
a = -10
Or
(a,b,c) = (-9-c,6+c,c)
(a,b,c) = (-9-1,6+1,1)
(a,b,c) = (-10,7,1)
Therefore,
------------------------------------------------------------
One more example
Let's say c = 2
b-c = 6
b-2 = 6
b = 6+2
b = 8
Then,
a+b = -3
a+8 = -3
a = -3-8
a = -11
Or
(a,b,c) = (-9-c,6+c,c)
(a,b,c) = (-9-2,6+2,2)
(a,b,c) = (-11,8,2)
Therefore,
It appears we keep landing on 54.
Is this a coincidence? Or is this always going to happen?
The next section will shed light on that.
------------------------------------------------------------
A more generalized approach.
b-c = 6
b = 6+c
a+b = -3
a+(6+c) = -3
a = -3-6-c
a = -9-c
We have
a = -9-c
b = 6+c
c = c
in which we can say
(a,b,c) = (-9-c,6+c,c)
This confirms that the system a+b = -3 and b-c = 6 has infinitely many solutions.
So,
This proves that if a+b = -3 and b-c = 6, then will always land on 54.
--------------------------------------------------------------------------
--------------------------------------------------------------------------
Answer: 54
Answer by greenestamps(13327) (Show Source):
Question 1205373: The perimeter of a rectangle is 48cm. The length of a rectangle is 3cm longer than twice its width. Find the dimensions of the rectangle.
Answer by math_tutor2020(3835) (Show Source):
You can put this solution on YOUR website!
Answer: 17 cm by 7 cm
Work Shown
w = width
2w+3 = length
2*(length + width) = perimeter of rectangle
2*(2w+3 + w) = 48
2*(3w+3) = 48
6w+6 = 48
6w = 48-6
6w = 42
w = 42/6
w = 7 cm is the width
2w+3 = 2*7+3 = 17 cm is the length
Check:
perimeter = 2*(length+width) = 2*(17+7) = 2*24 = 48 cm
We have confirmed the answers.
Question 1205305: A farmer has fenced off his trapezoidal housing area, which is shown in the diagram. There is a post at each of the point A and B, to which the farmer sometimes attaches a 30m rope that is tied to his donkey. This provides the donkey either one of two grazing areas outside the housing area. Find the difference in the areas available for grazing in square meters (m^2)
Found 2 solutions by math_tutor2020, Edwin McCravy:
Answer by math_tutor2020(3835) (Show Source):
Answer by Edwin McCravy(20077) (Show Source):
Question 1205272: Abu compares package postpaid mobile plans between Remaja Hebat Plan and
Unlimited Plan. Remaja Hebat Plan charges RM0.05 per minutes plus a basic
monthly fee for RM 7.50. Unlimited Plan charges RM 0.075 per minute plus a basic
monthly fee of RM4.25
a. Write the model for package postpaid mobile plans for Remaja Hebat Plan
and Unlimited Plan.
b. Identify the best mobile plan.
c. Sketch the graph for package postpaid mobile plans between Remaja Hebat
Plan and Unlimited Plan
. Alia rows a boat upstream from one point on a river to another point 4 km away in 1.5 hours.
The return trip, traveling with the current, takes only 45 min.
a. Identify the variables involve in algebra.
b. Find the speed of the current flowing by using back-substitution method
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
Do not pack more than 1 (one) problem per post.
It is against the rules of this forum; against the common sense,
perpendicular to the rules of good behavior and against your own interests.
For the rules of this forum, read at this web-page
https://www.algebra.com/tutors/students/ask.mpl?action=ask_question&topic=Equations&return_url=http://www.algebra.com/algebra/homework/equations/
from which you post your problems. Read it attentively and find many useful instructions there.
Become a civilized user / visitor . . .
Question 1204788: There were 1200 persons at a crusade, 8004 of them were women of the remaining persons persons,there were 3 times as many children as men. How many children were there?
Found 2 solutions by mananth, MathLover1:
Answer by mananth(16949)  (Show Source):
You can put this solution on YOUR website! There were 1200 persons at a crusade, 8004 of them were women of the remaining persons,there were 3 times as many children as men. How many children were there?
There is a typing error check and repost
Answer by MathLover1(20855) (Show Source):
Question 1204616: For a cylinder with a surface area of 10
, what is the maximum volume that it can have? Round your answer to the nearest 4 decimal places.
Recall that the volume of a cylinder is πr2h
and the surface area is 2πrh+2πr2
where r
is the radius and h
is the height.
Answer by math_tutor2020(3835) (Show Source):
Question 1204598: System of equations:
-7x+3y=-65
-7x+10y=-135
Write an equation that results from subtracting two equations
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
System of equations:
-7x+3y=-65
-7x+10y=-135
Write an equation that results from subtracting two equations
~~~~~~~~~~~~~~~~~~~~~~~~~
If you subtract first equation from the second equation, the resulting equation will be
10y - 3y = -135 - (-65),
or
7y = -135 + 65,
or
7y = -70 <<<---=== final
Subtracted and explained.
Question 1204043: The sum of two numbers is -22. The difference of the two numbers is 8. What are the  numbers?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13327) (Show Source):
You can put this solution on YOUR website!
The solution shown by tutor @ikleyn is probably the easiest, most straightforward way to solve the problem using formal algebra.
Of the many other ways to solve the problem that she mentions, there is one that I find to be a quick and easy informal method, using logical reasoning and simple mental arithmetic.
We are given that the sum of two numbers is -22 and the difference is 8.
If we picture that on a number line, then we start at the first number and go a distance equal to the second number in one direction to end up at -22, and we start at the first number and go a distance equal to the second number in the other direction to end up at 8.
That means the first number is halfway between -22 and 8.
So the first number is the average of -22 and 8; and then the second number is the difference between that average and either -22 or +8.
First number: (-22+8)/2 = -7
Second number: (-22)-(-7) = -15 (or (-7)-(8) = -15)
ANSWERS: -7 and -15
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
x + y = -22
x - y = 8
------------------- Add these equations to get
2x = -22 + 8 = -14 ---> x = -14/2 = -7; y = -22 - x = -22 - (-7) = -22 + 7 = -15.
ANSWER. The numbers are -7 and -15.
Solved.
There are other ways to solve, but this one is the most educative (in my view).
Question 1204044: The sum of two numbers is -22. The difference of the two numbers is 8. What are the two numbers?
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
x + y = -22
x - y = 8
------------------- Add these equations to get
2x = -22 + 8 = -14 ---> x = -14/2 = -7; y = -22 - x = -22 - (-7) = -22 + 7 = -15.
ANSWER. The numbers are -7 and -15.
Solved.
There are other ways to solve, but this one is the most educative (in my view).
Question 1202975: A wire 80 cm in length is cut into two parts and each part is bent to form a square. If the sum of the areas of the squares is 300 cm?, find the lengths of the sides of the two squares
Answer by math_helper(2461)  (Show Source):
Question 1202774: In what ways can you change a $100 bill using $10 bills, $5 bills, and $1 bills?
Answer by greenestamps(13327) (Show Source):
You can put this solution on YOUR website!
Whether you really want a list of all the different ways, or whether you only want to know HOW MANY ways there are, I will show you HOW to solve the problem and leave the details to you.
(1) Since the $10 bills and $5 bills together make a total number of dollars that is a multiple of 5, and since the total $100 is a multiple of 5, the total value of the $1 bills must be a multiple of 5.
(2) For each number of $1 bills that is a multiple of 5, the number of ways to make change for $100 is determined by the number of $10 bills you can have. For example, if there are 55 $1 bills, then the remaining amount is $45, and you can use 0, 1, 2, 3, or 4 $10 bills, making up the rest with $5 bills. That makes 5 ways to make change for $100 if the number of $1 bills is 55.
To outline the complete solution, then....
column 1: # of $1 bills
column 2: remaining amount
column 3: # of choices for the number of $10 bills
0 100 11 (0 to a maximum of 10)
5 95 10 (0 to a maximum of 9)
10 90 10 (0 to a maximum of 9)
...
...
85 15 2 (0 to a maximum of 1)
90 10 2 (0 to a maximum of 1)
95 5 1 (only 0)
100 0 1 (only 0)
Add the numbers in column 3 to find the total number of ways of making change for $100 using $10, $5, and $1 bills.
Note there is a nice pattern in the numbers in column 3 that makes it possible to find the total without adding the numbers one at a time....
Question 1202599: Write an exponential model for the following situation. The drug dosage is 375 mg. The drug is eliminated at a rate of 11.3% per hour. Use D=the amount of the drug in milligrams and t=time in hours. Enter your model in the simplified form y=asup(bracket(b),x), and be mindful about the case of your variables.
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(53748) (Show Source):
You can put this solution on YOUR website! .
Write an exponential model for the following situation.
The drug dosage is 375 mg. The drug is eliminated at a rate of 11.3% per hour.
Use D=the amount of the drug in milligrams and t=time in hours.
Enter your model in the simplified form y=asup(bracket(b),x),
and be mindful about the case of your variables.
~~~~~~~~~~~~~~~~~~
Below I present for you a STANDARD mantra on how to solve such problems.
The starting amount of the drug is 375 mg, and the exponential rate of eliminating the drug
is 11.3% per hour. It means that in terms of (D,t) the exponential model is
D =  =  . (1)
Here 375 is the initial/starting amount, given in this problem;
1-0.113 = 0.887 is the reducing factor per hour; t is the time, in hours.
+-----------------------------------------------------------------+
| Again, knowing the starting amount and the exponential rate |
| is just ENOUGH to write an exponential model (1) in whole. |
+-----------------------------------------------------------------+
In (y,x) form, formula (1) becomes
y =  .
That is all the mantra.
You do not need to pronounce any more words, because excessive words are UNNECESSARY and IRRELEVANT.
Moreover, if you will pronounce excessive words, everybody around will understand immediately
that you do not know the subject and that nobody and never did explain the subject to you in a right way.
-------------------
To see many other similar and different solved problems, look into the lesson
- A medication decay in a human's body
in this site.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the drug dose is 375 milligrams.
the drug is eliminated at a rate of 11.3% per hour.\
D = amount of drug in milligrams.
t = time in hours.
the equation i use most for these if f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the growth rate per time period
(1 + r) is the growth factor per time period.
n is the number of time periods.
in your problem:
p would be equal to 375 milligrams.
r would be -.113 per hour.
(1 + r) would be (1 - .113) = .887 per hour.
n would be the number of hours.
the formula would be simplified to f = 375 * .887 ^ n
replacing f with D and n with t, the formula would becomes D = 375 * .887 ^ t.
this would be you solution to this problem.
if i graph that formula, i would replace D with y and t with x to get y = 375 * .887 ^ x
here's what the graph would look like.
the amount of drug after 1 hours would be D = 375 * .887 ^ 1 = 332.675, as shown in the graph.
the half life of the drug would be calculated as follows:
D = 375 * .887 ^ t becomes 187.5 = 375 * .887 ^ t.
divide both sides of the equation by 187.5 to get .5 = .887 ^ t.
take the log of both sides of the equation to get:
log(.5) = log(.887 ^ t)
by log rule that says log(x^t) equals t * log(x), the formula becomes log(.5) = t * log(.887).
divide both sides of the equation by log(.887) to get log(.5) / log(.887) = t
solve for t to get t = 5.780547624.
round to 3 decimal placdes to get t = 5.781
this agrees with what's on the graph.
don't forget that D is the same as y on the graph and t is the same as x on the graph.
i don't know what y=asup(bracket(b),x), stands for.
a form of the exponential equation would be y = ab^x
if that's the case, then a would be 375 and b would be .887.
y = ab^x would then be y = 375 * .887 ^ x
this is the same as D = 375 * .887 ^ t, with y representing D and x representing t.
let me know if you have any questions.
theo
Question 1202419: There was $2900 in the pot. If there were 293 more $1 bills than $10 bills, how many bills of each kind were there?
Answer by josgarithmetic(39792) (Show Source):
|
Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825, 3826..3870, 3871..3915
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