Chapter 7, Arithmetic Review Problems
with hints of Algebra
Volume 2, Three Skills for Algebra
Here are some arithmetic review problems with hints of algebra. I would
give these problems at the start of a calculus or precalculus course to
check skills and correct common mistakes. Further, some arithmetic
patterns below provide experience with geometric sums and quick,
algebraic summation, formulas.
There is more to mathematics than just doing arithmetic carefully, but
in arithmetic you must master addition of fractions with least common
denominators and the cancellation of common divisors in fractions by
themselves or in products to do well in algebra and in calculus.
Calculus requires algebra and exact arithmetic with great strength and
precision. Answers or solutions provide the correct answers and
format for those answers.
2. Arithmetic and Algebra Review Problems
This large set of arithmetic and algebra review problems may help
you to check or diagnose your arithmetic skills and some algebra skills
as well - if you have studied algebra before. Answers will be found at
the end of this book. Doing these problems and seeing their answers
checks for gaps in your understanding, and may even fill some. Watch for
arithmetic or algebraic patterns in the last of these "review" problems.
Some calculations are slightly repetitive to help you spot such patterns.
2.1 Basic Stuff
Perform the indicated calculations by hand. Then check your calculations
with the aid of a calculator.
- Find the sum of the three numbers 456 and 76 and 312.
- Find the product of 176 and 86.
- Subtract 2396 from 4892 and check your answer.
- Compute 1416 divided by 813 to 3 decimal places.
- Compute 2396 -4892.
2.2 More Basic Stuff
Compute if possible the value of the following
Remember in calculations that operations inside parentheses $( )$ or
brackets [ ] are to be done first. Use your calculator as little as
possible.
-
$A = (4 \div 5)\div 3$
-
$B = 4 \div \frac53$
-
$C = 4 \times (5 \times 3)$
-
$ D = (4 \times 5) \times 3$
-
$E = (4 - 5) - 3$
-
$F = 4 - (5 - 3)$
-
$G = 4 - 5 -3$
-
$H = \sqrt{3^2}$
-
$I = \sqrt{(-3)^2}$
-
$J = \sqrt{ 4^2}$
-
$K = \sqrt{ (4^2+3^2)}$
-
$L = \sqrt{ (4^2 + (-3)^2)}$
-
$ M = (\frac54) \div [ (\frac87)\div (\frac95) ]$
-
$N = [(\frac54) \div (\frac87)] \div (\frac95)$
-
$O = \frac{5}{4} \times [ \frac78 \times \frac95 ]$
-
$P = [\frac{5}{4} \times \frac78] \times \frac95$
-
$Q = \frac{5}{4} \div \frac78 \div \frac95 $
-
$R = \sqrt{16} + \sqrt{9} - \sqrt{25}$
-
$S = (3.1416)^0$
-
$T = 3.1416 - \frac{22}7$
-
$U = \pi - 3.1416$
-
$V = \sqrt{4^2-5^2}$
2.3 Calculator Button Exercises
Put your calculator in degree mode. Now find or compute the following
quantities.
-
$A=\sin (90 ^\circ)$
-
$B=\sin (180 ^\circ)$
-
$C=\sin (0 ^\circ)$
-
$D=\sin (270 ^\circ)$
-
$E=\sin (-90 ^\circ)$
-
$F=\sin (-720 ^\circ)$
-
$G=\cos (90 ^\circ)$
-
$H=\cos (180 ^\circ)$
-
$I=\cos (360 ^\circ)$
-
$J=\cos (0 ^\circ)$
-
$K=\cos (-90 ^\circ)$
-
$L=\cos (-720 ^\circ)$
Put your calculator in radian mode. Now find or compute:
-
$a=\sin (\frac12\pi \mbox{ radians})$
-
$b=\sin ( \pi \mbox{ radians}) $
-
$c=\sin (0 \mbox{ radians})$
-
$d=\sin (\frac32\pi \mbox{ radians})$
-
$e=\sin (-\frac12\pi \mbox{ radians})$
-
$f=\sin (-4\pi \mbox{ radians})$
-
$g=\cos (\frac12\pi \mbox{ radians})$
-
$f=\cos (\pi \mbox{ radians})$
-
$h=\cos (2\pi \mbox{ radians})$
-
$i=\cos (1.5 \pi \mbox{ radians})$
-
$j=\cos (-\frac12\pi \mbox{ radians})$
-
$k=\cos (-4\pi \mbox{ radians})$
Observe that the numerical values computed by the sine,
cosine, tangent and all other trig-related function
buttons, all depends on the units used for angle measurement.
2.4 More Calculator Button Work
Compute or find the following quantities:
-
$ A= \exp( 2 \ln(5))$
-
$B=e^{2 \ln(5)}$
-
$ C= 10^{ 2 \log(5) }$
-
$ D= 10^{\log(25)}$
-
$ E= \ln(\exp(6.2))$
-
$F= \ln( e^{6.2})$
-
$G,$ the sixth root of $(16)^{12}$
-
$ H= \left[(16)^{12}\right]^{\frac16}$
-
$ I= 1+3+3^2+3^3+3^4+3^5+3^7$
-
$ J= \frac{-1+3^7}{[-1+3]}$
-
$ K= \frac{[1-3^7]}{[1-3]}$
-
$ M= 1+(1.06)+(1.06)^2+(1.06)^3$
-
$ N= \frac{[-1+(1.06)^4]}{[-1+1.06]}$
-
$ P= \frac{[(1.06)^4-1]}{[1.06-1]}$
-
$ Q= \frac{[-1+(1.06)^4]}{[0.06]}$
-
$R= [1+(1.02)^{1}+(1.02)^{2}+(1.02)^{3}+ (1.02)^{4}]\times
(1.02)^{(-4)}$
-
$ S= \frac{[(1.02)^{(5)}-1]}{[1.02-1]} \times (1.02)^{(-4)}$
-
$ T= 1+(1.02)^{(-1)}+(1.02)^{(-2)} +(1.02)^{(-3)}+(1.02)^{(-4)}$
-
$ U= (1.02)^{(-4)}+(1.02)^{(-3)}+(1.02)^{(-2)}+(1.02)^{(-1)}+1$
-
$V= \frac{ (\frac1{1.02})^{(5)}-1}{(\frac1{1.02})^{(1)}-1}$
2.5 More Arithmetic Examples
Answer the following without the use of a calculator - One of the
following is not defined. See Answers.
- Simplify, if defined, \[A= [(\frac4{5}) \div (\frac{24}{35})]
\div (\frac2{7})\]
- Simplify, if defined, \[B=(\frac4{5}) \div [(\frac{24}{35})
\div (\frac2{7})]\]
- Simplify, if defined, \[ C=(\frac4{5}) \div (\frac{24}{35})
\div (\frac2{7}) \]
2.6 A Summation Shortcut
Before the shortcut is given, we will tackle two suggestive tasks. The
sum of the cubes of the integers 1 to 4 is S =
1+23+33+43. Your first task is to
compute S. Your second task is to compute
\[a=[(\frac12)4(4+1)]^2\]
Now compare the values of big S and little a.
Now here is the shortcut: if n is a positive integers then the sum
of the cubes of the integers 1 to n is \[ S(n)=[\frac12n(n+1)]^2
\]
Why it holds is an intellectual debt: It can be justified or proven with
the help of mathematical induction and the ideas in the chapter Some
Finite Mathematics. With this formula, and also without if you like,
find
- the sum of the cubes of the integers 1 to 5
- the sum of the cubes of the integers 1 to 15
- the sum of the cubes of the integers 1 to 30
In the last problem, what requires the least amount of arithmetic, use
of the formula $S(n)=[\frac12n(n+1)]^2$
or directly adding the 30 cubes
$ 1^3, 2^3, 3^3, \ldots
(29)^3, (30)^3?$
2.7 Algebraic Exercises
Some may be harder than the previous ones. If the algebraic exercises are
not understandable now, try them later.
-
Simplify $\frac{1+x+x^2+x^3}{x-1}$ if possible.
-
Factor $x^2+5x+6,$ Hint: try to use the algebraic pattern $(x+a)(x+b)
= x^2+(a+b)x+ab$ with $ab=6,$
-
Solve $0=(x-1)(2x+4)(3-x),$ Hint: There are three numbers in the
answer.
-
Factor $x^3-x,$
-
Simplify $4(x+1)(x+3x^2)-[(x+1)x+(x+1)3x^2)],$
-
Simplify $13^2-5^2-(13+5)(13-5),$
-
Simplify $7- \sqrt{(3^2+4^2)},$
-
Simplify $[\left(\frac37\right)^{13} \times \left( \frac{(4x^2)}{(3^2
\times 7^3)}\right)^5],$
-
Simplify $[(9x^2+3)(4+4x+4x^2)][(x-1)(2x+2)-2x^2+2)],$
-
Compute $f(4)$ if $f(x)=\sqrt{25-x^2}.$
-
Find: $ (x,y)$ if $x+y =\pi$ and $y-x=1.$
-
Express $$[(2 \times 3^2\times y^3z^{(-3)}t^3)^{(-2)}] \times
[3^3x^4y^{(-5)}]^2$$ with positive powers only.
-
Find: $ x$ if $(x-10)(x-3) = 0$ and $ x > 4,$ Also, how many
acceptable solutions would there be if the requirement $x>4$ was
replaced by $x >12,$ or $x > 2$ or $x > -99$? Consider the
three alternate cases separately.
-
Find $ x$ if $4 =\frac1{(x+1)}$
-
Find $ z$ if $z=2x+3,$ $t=3^2,$ $w-t=10$ and $x= 4t+1$ and $y=y^2,$
Observe some information is not needed.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
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if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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