SAT Number Line

To solve problems about number lines on the SAT, you need to know:

• A number's distance from zero is its absolute value, and it is always positive.
• If \(a\) and \(b\) are on a number line, and \(a<b\), then the distance between \(a\) and \(b\) is \(b-a\).
• properties of numbers between \(0\) and \(1\)

For \(0<a<1\):

• If \(b\) is positive, then \(ba<a\).
• If \(m\) and \(n\) are integers, such that \(1<n<m\), then \(x<x^{n}<x^{m}\).
• \(a<\sqrt{a}\).
• \(a^{2}<a\).
• \(a<1<\frac{1}{a}\).

• ratios and proportions
• properties of inequalities

A number line is a graphical representation of the real numbers. It is a horizontal line on which numbers are represented as points, placed at equal intervals. Zero is located in the middle, the positive numbers to the right of zero, and the negative numbers to the left of zero. As you move to the right, numbers increase, and as you move to the left, numbers decrease. The number line extends infinitely on both sides, but usually only a part of it is drawn.

On the number line above, marks are equally spaced. Which of the following represents the length of one segment?

(A) \(\ \ 1\)

(B) \(\ \ \frac{1}{5}\)

(C) \(\ \ \frac{b-a}{5}\)

(D) \(\ \ b-a\)

(E) \(\ \ 5\)

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Correct Answer: C

Solution:

Since the marks are equally spaced, the lengths of the segments are the same. From the diagram we see that the distance between \(a\) and \(b\) is divided into \(5\) segments. Therefore,

\(\text{length of one segment} = \frac{b-a}{5}\)

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Incorrect Choices:

(A)
Tip: Only assume that the tick marks are equally spaced, nothing more.
It is common to see a number line that represent integers. The tick marks on such a number line are usually spaced \(1\) unit apart because consecutive integers differ by \(1\). If you assume that the distance between each tick mark is \(1\), you will be wrong. Here is a counter-example: If \(a=10\) and \(b=20\), then the length of each segment would be \(\frac{20-10}{5}=\frac{10}{5}=2\), not \(1\).

(B)
Tip: Look for a counter-example.
If you assume that the distance between \(a\) and \(b\) is \(1\), you will get this wrong answer. Here is a counter-example: If \(a=10\) and \(b=20\), then the length of each segment would be \(\frac{20-10}{5}=\frac{10}{5}=2\), not \(\frac{1}{5}\).

(D)
Tip: Read the entire question carefully.
If you solve for the distance between \(a\) and \(b\), you will get this wrong answer.

(E)
Tip: Read the entire question carefully.
Tip: Look for a counter-example.
If you solve for the number of segments between \(a\) and \(b\), you will get this wrong answer. Here is a counter-example: If \(a=10\) and \(b=20\), then the length of each segment would be \(\frac{20-10}{5}=\frac{10}{5}=2\), not \(5\).

On the number line above, the tick marks are equally spaced. What is the coordinate of point \(A\)?

(A) \(\ \ 0.7727\)
(B) \(\ \ 0.7729\)
(C) \(\ \ 0.7733\)
(D) \(\ \ 0.7737\)
(E) \(\ \ 0.7744\)

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Correct Answer: C

Solution:

Tip: Only assume that the tick marks are equally spaced, nothing more.
The distance between \(0.7745\) and \(0.7725\) is divided into five equal segments. Therefore,

\(\text{length of one segment}==\frac{0.7745-0.7725}{5}=\frac{0.0020}{5}=0.0004\).

Hence, the distance between adjacent tick marks is \(0.0004\). Point \(A\) is located two tick marks to the right of \(0.7725\). So,

\(A=0.7725+2\cdot 0.0004 = 0.7725+0.0008=0.7733\)

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Incorrect Choices:

(A)
Tip: Read diagrams carefully.
If you assume that the distance between adjacent tick marks is \(0.0001\), you will get this wrong answer. If tick marks were \(0.0001\) apart, then if we start at \(0.7725\), and move five tick marks to the right, we would end at \(0.7730\), not at \(0.7745\).

(B)
Tip: Read diagrams carefully.
If you assume that the distance between adjacent tick marks is \(0.0002\), you will get this wrong answer. If tick marks were \(0.0002\) apart, then if we start at \(0.7725\), and move five tick marks to the right, we would end at \(0.7735\), not at \(0.7745\).

(D)
\(A\) is located two tick marks to the right of \(0.7725\), not two tick marks to the left of \(0.7745\). If you do this \(0.7745 - 2\times 0.004=0.7737\), you will get this wrong answer.

(E)
Tip: Eliminate obviously wrong answers.
Tip: Estimate.
Trust a diagram that is drawn to scale. \(A\) is closer to \(0.7725\) than it is to \(0.7745\). If \(A\) were located at \(0.7744\), you would expect it to be much closer to \(0.7745\) than to \(0.7725\). You can reasonably eliminate this choice.

On the number line above, the ratio of \(AE\) to \(AG\) is equal to the ratio of \(DF\) to which of the following?

(A) \(\ \ AB\)
(B) \(\ \ AC\)
(C) \(\ \ AG\)
(D) \(\ \ BE\)
(E) \(\ \ EF\)

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Correct Answer: D

Solution 1:

Tip: Plug and check.
The length of \(AE\) is \(4\). The length of \(AG\) is \(6\). We find the ratio of \(AE\) to \(AG\):

\(\frac{AE}{AG}=\frac{4}{6}=\frac{2}{3}\).

The length of \(DF\) is \(2\). We find the ratio of \(DF\) to each of the answer choices, and select the option which yields \(\frac{2}{3}\).

(A) \(AB\) has length \(1\). \(\frac{DF}{AB}=\frac{2}{1} = 2\neq \frac{2}{3}\). Eliminate (A).
(B) \(AC\) has length \(2\). \(\frac{DF}{AC}=\frac{2}{2} = 1 \neq \frac{2}{3}\). Eliminate (B).
(C) \(AG\) has length \(6\). \(\frac{DF}{AG}=\frac{2}{6} = \frac{1}{3} \neq \frac{2}{3}\). Eliminate (B).
(D) \(BE\) has length \(3\). \(\frac{DF}{BE}=\frac{2}{3} = \frac{2}{3}\). This choice is correct.
(E) \(EF\) has length \(1\). \(\frac{DF}{EF}=\frac{2}{1} = 2\neq \frac{2}{3}\). Eliminate (E).

Solution 2:

The length of \(AE\) is \(4\). The length of \(AG\) is \(6\). Therefore, the ratio of \(AE\) to \(AG\) is \(4\) to \(6\).

The length of \(DF\) is \(2\). We know that the ratio of \(AE\) to \(AG\) is equal to the ratio of \(DF\) and another segment of unknown length. Let's call that length \(x\).

We set up a proportion and solve for \(x\):

\[\begin{array}{l c l l l} \frac{AE}{AG} &=& \frac{DF}{x} &\quad \text{set up a proportion} &(1)\\ \frac{4}{6} &=& \frac{2}{x} &\quad AE = 4, AG = 6, DF = 2 &(2)\\ x \cdot 4 &=& 2 \cdot 6 &\quad \text{cross-multiply} &(3)\\ 4x &=& 12 &\quad 2\cdot 6 = 12 &(4)\\ x &=& 3 &\quad \text{divide both sides by}\ 4 &(5)\\ \end{array}\]

The length of the unknown segment must be \(3\). Of the choices, only \(BE=3\).

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Incorrect Choices:

(A), (B), (C), and (E)
Refer to Solution 1 to see how to eliminate these choices.

If you thought these examples difficult and you need to review the material, these links will help:

• Number Line
• SAT Fractions and Decimals
• Absolute Value
• SAT Reasoning Skills

• Only assume that the tick marks are equally spaced, nothing more.
• Estimate.
• Read diagrams carefully.
• SAT General Tips