SAT Math Concise Study Notes : Numbers and Arithmetic

SAT MAth and English  – full syllabus practice tests

  • Algebra Weightage: 35%  Questions: 13-15
  • Advanced Math Weightage: 35% Questions: 13-15
  • Problem-solving and Data Analysis Weightage: 15%  Questions: 5-7
  • Geometry and Trigonometry Weightage: 15% Questions: 5-7

Integers are whole numbers, negative and positive. Zero is an integer, but is neither positive nor negative. Rational numbers are numbers that can be expressed as fractions. Real numbers include integers, rational numbers, and irrational numbers such as \(\pi, e\), and \(\sqrt{2}\).

  • Consecutive integers are integers in a sequence, each being 1 more than the previous integer:

$
n,(n+1),(n+2) \ldots
$

  • If the sum of three consecutive integers is greater than 84 , what is the smallest possible value of the first integer?

$
\begin{gathered}
n+(n+1)+(n+2)>84 \\
3 n+3>84 \\
n>27
\end{gathered}
$

The smallest possible value of \(n\) is 28 .
The sum of the consecutive integers from 1 to an integer \(n\) is equal to

$
(n+1) \times n / 2
$

  •  Take the set of consecutive integers from 1 to 30. The sum of the first and last integers is \(1+30=31\), the sum of the second and second from last integers is \(2+29=31\), and the sum of the third and third from last integers is \(3+28=31\). There will be \(30 / 2\) pairs of integers that each add to 31 , so the total sum will be \(31 \times 30 / 2=465\).

Adding or multiplying even or odd integers will result in a sum or product that is predictably even or odd:

  • even + even \(=\) even
  •  odd + odd = even
  • odd + even = odd
  • even \(\times\) even \(=\) even
  • odd \(\times\) odd \(=\) odd
  • odd \(\times\) even \(=\) even

An integer can be divided into a limited set of factors; there are an infinite number of multiples that can be divided by a specific integer. An integer with only 2 factors (itself and 1) is a prime
number. 1 is not a prime number, but 2 is.
The least common multiple of two integers is the smallest positive integer that is a multiple of both integers. The greatest common factor of two integers is the largest integer that is a factor of both integers. You can determine both of these by finding the prime factors of each integer.

  •  What is the difference between the least common multiple
    and the greatest common factor of 100 and 80?
  • Prime factors of \(100: 2 \times 2 \times 5 \times 5\)

    Prime factors of 80: \(2 \times 2 \times 2 \times 2 \times 5\)

100 and 80 share two \(2 \mathrm{~s}\) and one 5 . Multiplying these prime factors together gives us their greatest common factor: \(2 \times 2 \times 5=20\).

In total, four \(2 \mathrm{~s}\) and two \(5 \mathrm{~s}\) appear in these prime factorizations, excluding repeats. Multiplying these prime factors together gives us their least common multiple: \(2 \times 2 \times 2 \times 2\) \(\times 5 \times 5=400\).

The difference between the LCM and the GCF is \(400-20=380\).

 

 

 

A fraction generally expresses a part/whole relationship between two quantities. A ratio generally expresses a part/part relationship among two or more quantities.

  •  If the ratio of boys to girls in the class is 12 to \(17=12: 17=\frac{12}{17}\), then the fraction of boys in the entire class is \(\frac{12}{12+17}=\frac{12}{29}\).
  • If the ratio of green to blue to red marbles in the jar is 5:4:3, then the fraction of green marbles in the jar is \(\frac{5}{5+4+3}=\frac{5}{12}\).

A proportion is an equation where two ratios are set equal to each other. Solve proportions by cross-multiplying.
$
\text { If } \frac{a}{b}=\frac{c}{d} \text {, then } a d=b c
$

  •  A train has travelled 40 miles in the past 45 minutes. If the train is moving at a constant speed, what distance will it have travelled after 4 hours?

Convert to the same units:
Set up a proportion:
Cross-multiply and solve:
$
\begin{gathered}
45 \text { minutes }=0.75 \text { hours. } \\
\frac{40 \text { miles }}{0.75 \text { hours }}=\frac{x \text { miles }}{4 \text { hours }} \\
40 \times 4=0.75 x \\
0.75 x=160 \\
x=213 \frac{1}{3} \text { miles }
\end{gathered}
$

A percent can be represented as a fraction whose denominator is 100 or as a decimal: \(75 \%=\) \(75 / 100=0.75\). Solve a percent problem by converting the percent into a fraction and crossmultiplying:

  • 30 is \(60 \%\) of what number?
    $
    \begin{gathered}
    \frac{30}{x}=\frac{60}{100} \\
    3000=60 x, \text { so } x=50
    \end{gathered}
    $

Alternatively, you can set up an equation and convert the percent into a decimal. The word of means multiplication:

  •  30 is what percent of 150 ? \(30=x \times 150\), so \(x=0.2=20 \%\)
  •  30 is \(60 \%\) of what number? \(30=0.6 x\), so \(x=50\)

Percent increase or percent decrease is equal to
$
\frac{\text { change in amount }}{\text { original amount }} \times 100 \%
$

  • The price of apples has decreased from \(\$ 0.99 / \mathrm{lb}\) to \(\$ 0.79 / \mathrm{lb}\). What is the percent decrease in price?

$
\frac{\text { change in amount }}{\text { original amount }} \times 100 \%=\frac{\$ 0.99-\$ 0.79}{\$ 0.99} \times 100 \%=20.2 \%
$

  •  A book has been discounted \(15 \%\) and its current price is \(\$ 12\). What was its original price?

$
\begin{gathered}
\frac{x-\$ 12}{x} \times 100 \%=15 \% \\
x-\$ 12=0.15 x \\
0.85 x=\$ 12 \\
x=\$ 14.12
\end{gathered}
$

Exponent notation indicates that a number is being multiplied by itself. The number is the base and the number of times it is being multiplied is the exponent. Exponents follow standard rules:

  • If \(5^{\frac{x}{2}}=\sqrt{5^a \times 5^b}\), then what is \(x\) in terms of \(a\) and \(b\) ?

$
\begin{gathered}
5^{\frac{x}{2}}=\sqrt{5^x} \\
\sqrt{5^a \times 5^b}=\sqrt{5^{a+b}} \\
\sqrt{5^x}=\sqrt{5^{a+b}} \\
x=a+b
\end{gathered}
$

The absolute value of a number is its distance away from 0 on a number line. The absolute value will always be positive.

$
\begin{gathered}
\text { If }|x|=4 \text {, then } x=4 \text { or } x=-4 \\
\text { If }|x|<4 \text {, then }-4<x<4 \\
\text { If }|x|>4 \text {, then } x<-4 \text { or } x>4
\end{gathered}
$

  • Solve:

$
\begin{gathered}
|x-5|=4 \\
x-5=4 \text { or } x-5=-4 \\
x=9 \text { or } x=1
\end{gathered}
$

This means \(x\) is 4 units away from 5 in either the positive or negative direction.

  • A manufacturer of cereal will discard all boxes weighing less than \(28.5 \mathrm{oz}\) and more than \(31.5 \mathrm{oz}\). What absolute value equation represents all weights \(x\) that will be discarded?

\(28.5 \mathrm{oz}\) and \(31.5 \mathrm{oz}\) are both \(1.5 \mathrm{oz}\) away from \(30 \mathrm{oz}\). If these were the weights that were discarded, we would write
Test:
$
\begin{gathered}
|x-30|=1.5 \\
x-30=1.5 \text { or } x-30=-1.5 \\
x=31.5 \text { or } x=28.5
\end{gathered}
$

However, the manufacturer will discard all boxes that weigh less than \(28.5 \mathrm{oz}\) and more than \(31.5 \mathrm{oz}-\) that is, those that differ more than \(1.5 \mathrm{oz}\) from the standard weight of \(30 \mathrm{oz}\). Our equation should read

Test:
$
\begin{gathered}
|x-30|>1.5 \\
x-30>1.5 \text { or } x-30<-1.5 \\
x>31.5 \text { or } x<28.5
\end{gathered}
$

A set is an unordered collection of items. These can be numbers, colors, letters, days of the week, other sets, etc. The union of two sets is a set consisting of all of the elements of both sets. The intersection of two sets is a set consisting of only the shared elements.

  •  If \(A\) is the set of the first three positive even integers and set \(B=\{1,2,3,4\}\), what is the union of the two sets? What is the intersection of the two sets?

$
A=\{2,4,6\} \text { and } B=\{1,2,3,4\} \text {, so their union is }\{1,2,3,4,6\} \text {. Their intersection is }\{2,4\} \text {. }
$

A sequence is an ordered list of numbers, often following a specific pattern. A sequence can be definite or indefinite. In an arithmetic sequence, the next term is created by adding a constant to the previous term. In a geometric sequence, the next term is created by multiplying a constant to the previous term.

  •  Arithmetic sequence: \(2,6,10,14,18,22 \ldots\)
  •  Geometric sequence: \(3,12,48,192,768 \ldots\)

You may need to find the sum or average of certain terms in a sequence, or the value of a specific term in a sequence. To answer any sequence question, always write out at least 5 terms to establish

  • the pattern. You will never be asked to derive the formula for a sequence, but you may need to do something of the sort in order to answer the questions. Keep in mind:

 If \(a_1\) is the first term of an arithmetic sequence, \(d\) is the common difference between terms, and \(n\) is the number of terms, then
$
a_n=a_1+d(n-1)
$

  •  If \(a_1\) is the first term of a geometric sequence, \(r\) is the common ratio between terms, and \(n\) is the number of terms, then

$
a_n=a_1(r)^{n-1}
$

  • Each term in a sequence is 3 times the preceding term. If the first term is 2, what is the average of the \(5^{\text {th }}\) and \(7^{\text {th }}\) terms?

$
\begin{gathered}
a_7=2 \times 3^6=1458 \\
a_5=2 \times 3^4=162
\end{gathered}
$

The average of the two terms is
$
\frac{1458+162}{2}=810
$

Some sequences are neither arithmetic nor geometric. Establish a pattern, but don’t waste your time trying to find an algebraic formula for these!

  •  \(1,4,5,9,14,23,37\) : each term is the sum of the previous two terms.
  •  \(4,6,9,3,4,6,9,3\) : the terms repeat cyclically.
  • A lane divider in a swimming pool has 77 flags strung in a row. The colours repeat in a pattern: red, blue, green, red, blue, green, red, blue, green … If the first flag is red, what colour is the last flag?
    The three colours will repeat themselves \(77 / 3=25.66667\) times, or 25 times with two remainders. Following the pattern, these last two flags will be red and blue, respectively. Thus, the last (77th) flag is blue.
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