Download Combinations – Black Problems Blue Level Word

Combinations – Black Problems
Blue Level Word Problems for Reinforcement
1. Ann gave 1 of her beads to Joey, 3 of them to May and 1 of the rest to Rose.
4
16
5
If Ann had 36 beads left, how many beads did she have at first?
2. Henry has 3 as many paper – clips as Joyce. Joyce has 4 as many paper – clips as
4
5
Claire. If the three girls have 96 paper – clips altogether, how many fewer paper –
clips does Henry have than Claire?
3. John and Rick had 900 stamps altogether. After John used 2 of his stamps and
7
Rick used 180 of his stamps, they had the same number of stamps left. How many
stamps did they have left?
Blue Level Problems
4. Wendy saves 30% of her income each month. 3 of her expenditure is spent on
8
1
2
cosmetics, of it spent on transport and of the remainder is spent on food. If
6
5
she spends $469 on cosmetics and food altogether, how much does she save each
month?
5. Mabel has 5 as many stickers as Claire. Claire has 5 as many stickers as Sarah. If
8
6
the 3 girls have 226 stickers altogether, how many more stickers does Sarah have
than Mabel?
6. Mike has 2 as much money as Peter. Peter has 5 as much money as Irene. After
3
6
Irene bought 3 notebooks at $2.40 each, she had $8.80 more than Mike. How
much money did Peter have?
Black Level Problems
7. David bought 4 kg of vegetables, 3 kg of prawns and 1 kg of meat for $13.20
5
4
2
altogether. He bought the prawns at $10 per kg. If he had not bought the meat
but another 1 kg of vegetables instead, he would have spent $2.10 less. How
10
much did he pay for each kg of meat?
8. A man bought 3 sacks of flour at $16 per sack. Each sack of flour weighed 30 kg.
He repacked the flour into bags of 3 kg and bags of 1 1 kg. There were five times
2
5
3
1
kg bag at $0.50 and each 1 kg bag at $1.20, how much profit did he
as many
5
2
make in all?
9. In a class of 40 students, 3 of them play only basketball, 2 play only soccer and
8
5
3 play only volleyball. If 3 of those who play basketball and 1 of those who play
20
5
4
soccer now also play volleyball, how many students play volleyball altogether?
10. Thanksgiving Travel. Zach is planning to drive to his parents’ house in
Philadelphia for Thanksgiving. He lives in Pittsburgh so he can either take Route
30, which is a scenic drive, or take the Pennsylvania Turnpike. He’s trying to
decide between the two routes. Here are the details that he’s considering:
Pennsylvania Turnpike:
- distance is 305 miles each way
- toll road charge is $9.00 each way
1 of the trip is city driving and 60 is highway driving
61
61
Route 30:
- distance is 372 miles each way
- no tolls
- 1 of the trip is city driving and 2 is highway driving
3
3
Gas costs on the average $1.60 per gallon. Zach’s car gets 25 miles per gallon in
the city and 30 miles per gallon on the highway. (Route 30 involves more city
driving because he’d be driving through a lot of small towns.)
Explain which plan is the most economical (not considering the time involved or the
wear and tear on is car).
COMBINED OPERATIONS AND SPECIAL CASES
There are some special cases you should know about when you simplify rational expressions,
such as
5− x
x−5
Here, the numerator and denominator are opposites of each other, as you can see by
factoring -1 from the numerator.
5 − x −1(−5 + x) −1( x − 5)
=
=
= −1
x −5
x −5
x −5
The quick way to get this answer is to reason:
Any number divided by its opposite equals -1.
Caution is required here because some expressions look like opposites but really are not.
For instance,
x - 5 and x + 5
are conjugates, not opposites. Nothing special happens when you divide two conjugates.
SPECIAL CASES
x −5
= −1
5− x
A number divided by its opposite equals – 1.
x+5
=1
5+ x
A number divided by itself equals 1.
x −5 x −5
=
x+5 x+5
A number divided by its conjugate
(x + 5 and 5 + x are equal.)
is nothing special!
Another special case concerns the opposite of a fraction, such as
-7,
9
-7 ,
9
and
7 .
-9
The expression 7 means 7 divided by 9. Because negative divided by positive is negative
9
and positive divided by negative is also negative, all three of these fractions are equal.
- 7 = -7 = 7
9
9
-9
The – sign can be associated with the numerator, the denominator, or the entire fraction.
This fact is called the property of the opposite of a fraction.
PROPERTY
OPPOSITE OF A FRACTION
For any fraction a , b=0,
b
- a = -a = a .
b
b
-b
Note that - a equals + a , since negative divided by negative is positive.
-b
b
Objective
Be able to add, subtract, multiply, and divide rational algebraic expressions, and simplify
the answer.
Cover the answer as you work these examples.
EXAMPLE 1
Simplify:
( x + 2)( x − 3)( x − 4)
.
( x + 3)(4 − x)( x + 2)
-----
( x + 2)( x − 3)( x − 4)
( x + 3)(4 − x)( x + 2)
( x − 3)( x − 4)
( x + 3)(4 − x)
=−
x−3
x+3
EXAMPLE 2
Perform the operations and simplify:
----
Think These Reasons
Write the given expression.
Cancel the (x+2) factors.
( x − 4)
equals -1. (x-3) and (x+3) are
(4 − x)
conjugates and do not cancel.
2x 4
3
− ÷
3 x 5 x 10 x
-----
-----
2x 4
3
− ÷
3 x 5 x 10 x
Write the given expression.
2 x 4 10 x
− •
3x 5 x 3
Multiply by the reciprocal.
=
2 x 40 x
−
3 x 15 x
2 8
= 3 3
Multiply before subtracting.
Cancel the | factors. Cancel 5.
=- 6
3
=- 2
EXAMPLE 3
Subtract and simplify:
-----
Subtract the fractions.
Simplify.
3x − 1
2
−
x + 2 x − 15 x + 5
2
3x − 1
2
−
x + 2 x − 15 x + 5
2
3x − 1
2
−
( x + 5)( x − 3) x + 5
=
3x − 1
2
x −3
−
•
( x + 5)( x − 3) x + 5 x − 3
-----
Write the given expression.
Factor the denominator.
Write the fractions with common
denominators. (You need transform only the
=
3 x − 1 − 2( x − 3)
( x + 5)( x − 3)
second fraction.)
Subtract the numerators.
Use the common denominator.
3x − 1 − 2 x + 3
( x + 5)( x − 3)
Distribute -2. (Be careful not to lose
the – sign!)
x+5
( x + 5)( x − 3)
Combine like terms.
1
x−3
Cancel the ( x + 5) factors. Do not lose the
11.
2- 3 + 4
a
a2
a3
12.
6
5 3x 4
•
÷
x 2 2 x3 −4
13.
6u + 12
15u
•
5
11u + 22
14.
13 x
26
−
x−2 x−2
15.
7
4
+
x −3 3− x
1 in the numerator. Also, keep the
denominator in factored form until the very
end to see if any canceling can be done.
16.
2 x − 11
3
+
x − 7 x + 12 ( x − 4)
2
17.
5 x + 17
3
−
x + 8x + 7 x + 7
2
18. The Density Property. The diagram shows a number line. Between 3 and 4 there
are many other numbers such as 3.5, 3.79, 3.22861154. . . , etc. If a set of
numbers has the property that between any two of them there is another number
of that kind, then the set of numbers is said to be dense. The work you have been
doing with fractions allows you to tell whether or not certain sets of numbers are
dense.
There are real numbers between 3 and 4.
0
1
2
3
: : :4
Answer the following questions.
a. Find a real number between 4.8 and 4.9.
b. Find five real numbers between 7.83 and 7.84.
Combinations – Black Solutions
1
4
1.
3
16
?
1
5
36
After
giving
part
of
her
beads
= 1 - 1 - 3 = 16 - 4 - 3 = 9
4 16 16 16 16 16
to
Joey
and
May,
After giving part of her beads to Rose, fraction of Ann’s beads left
Number of beads Ann had at first
fraction
of
Ann’s
beads
9
⎛ 1⎞ 9 4 9
⎜1 − ⎟ • = • =
⎝ 5 ⎠ 16 5 16 20
9
20
20
= 36 •
= 80
9
= 36 ÷
She had 80 beads at first.
left
2.
Claire
96
Joyce
Henry
?
5 + 4 + 3 = 12 units
12 units -> 96 paper-clips
1 unit -> 96
÷
12= 8 paper-clips
5 – 3 = 2 units
2 units -> 2 x 8 =16 paper-clips
Henry has 16 fewer paper-clips than Claire.
Blue Level Solutions
3.
Before
John
900
Rick
180
Fraction of John’s stamps left
=1 - 2 = 5
7
7
After
John
?
Rick
1 unit -> 720
÷
7 + 5 = 12 units
12 units -> 900 – 180 = 720 stamps
12 = 60 stamps
5 + 5 = 10 units
10 units -> 10 x 60
= 600 stamps
They had 600 stamps left.
4.
$360
5.
46
6.
$30
Solutions to Black Level problems
7. $5
8.
$26
9.
10.
19
Thanksgiving Travel. Route 30 is more economical than the Pennsylvania Turnpike because Zach will only spend
$42.34 there and back instead of $50.64 there and back.
Zach should take Route 30 because he won’t spend as much money as he would on the Pennsylvania Turnpike. If
he took the Pennsylvania Turnpike he would spend $25.32 on gasoline and tolls instead of spending only $21.17 on
gasoline. This is how I found how much money each road trip cost.
Pennsylvania Turnpike:
First I found the amount of city miles
305
61
mi = 5 mi
Then I found the amount of gallons for the 5 miles. 25 mi per gallon for the city.
5 mi / 25 mi / gal
5 mi
25 mi
= . 2 gal
For the highway, I found the amount of miles first.
305 : 60
61
mi = 300 mi
Then I found the amount of gallons for the 300 miles. 30 mi per gallon for the highway.
300 mi / 30 mi / gal
300 mi
30 mi
= 10 gal
Finally I added the city’s gallons and the highway’s gallons together and multiplied the sum by the amount per
gallon. Then I added the toll price to the answer. Next I multiplied by 2, because the trip was there and back.
$16.32 + $9 (toll) = $25.32
. 2 + 10 = 10.2 gal
10.2 • $1.60 = $16.32
$25.32 • 2 = $50.64 (total cost)
Route 30:
First I found the amount of city miles
372
3
mi = 124 mi
Next I found the amount of gallons for the 124 miles. 25 mi per gallon for the city.
124 mi / 25 mi / gal
124 mi
25 mi
= 4.96 gal
For the highway, I found the amount of miles first.
372 mi : 2
3
= 248 mi
Then I found the amount of gallons for the 248 miles. 30 mi per gallon for the highway.
248 mi/ 30 mi / gal
248 mi
30 mi
= 8.27 gal
At last I added the city and highway’s gallons together and multiplied the sum by the amount per gallon. Then I
multiplied that answer by 2, since the trip was there and back.
4.96 + 8.27 = 13.23 gal
13.23 gal • $1.60 = $21.17
$21.17 • 2 = $42.34 (total cost)
Since $42.34 is less than $50.64, Zach should travel on Route 30.
11.
12.
2a 2 - 3a + 4
a3
−
20
x9
13.
18u
11
14.
13
15.
3
x−3
16.
5
x−3
17.
2
x +1
18.
a.
Answers may vary. 4.85 b. Answers may vary. 7.831, 7.833, 7.835