SHSAT Sunset

Which number shown above is the greatest

2024 Handbook

2024 Test B Math · Question 21 of 57

Question

−149, −49, −17, −1-\frac{1}{49},\ -49,\ -\frac{1}{7},\ -1

Which number shown above is the greatest?

  • A−149-\frac{1}{49}
  • B−49-49
  • C−17-\frac{1}{7}
  • D−1-1
Show the answer

Correct answer: A

All the numbers are negative, so they are all less than zero. Consider the absolute value of each number and where it falls on a number line. The negative number closest to 0 will be the greatest.

The negative number with the greatest absolute value is −49, so it has the least value.

The negative number with the next greatest absolute value is −1, so it would fall next in order from least to greatest.

−49, −1-49,\ -1

Compare the absolute values of −149-\frac{1}{49} and −17-\frac{1}{7}.

149<17\frac{1}{49}<\frac{1}{7}

Therefore, the numbers listed from least to greatest are:

−49, −1, −17, −149-49,\ -1,\ -\frac{1}{7},\ -\frac{1}{49}

So the number −149-\frac{1}{49} is the greatest.

Question analysis

Four negative numbers are listed: negative 1/49, negative 49, negative 1/7 and negative 1. Students decide which of them is the greatest. Comparing negative fractions with negative whole numbers tests whether students order numbers by their position on the number line or by their size without the sign.

NYS standards
  • 6.NS.7a inequality as relative position
  • 6.NS.7d absolute value versus order
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Ordering positive and negative rational numbers on a number line is grade 6 content in New York. This is a one-step comparison of four numbers, lighter than typical grade 7 work with signed numbers.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
2 steps
Reading level
Question and choices: about grade 10, 12 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

With no arithmetic to do, the question tests how firmly students picture negative numbers on a line. Comparing sizes without the sign picks the number farthest from 0, and comparing the two fractions by habit picks the wrong one of the pair. Placing all four as points left of 0, with −1 between −49 and the fractions, makes the order plain.

  1. A very close wrong answer moderate Choosing the answer

    −1/49 and −1/7 differ only in the denominator, and for negative numbers the larger denominator gives the greater value, the reverse of what their sizes suggest. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  2. Something you must already know minor Working it out

    Students must bring the number-line fact that a negative number's value falls as its distance from 0 grows, so −49 is the least of the four, not the greatest. (Anderson & Krathwohl, 2001)

Strategy: Picture each number's place on the number line: among negative numbers, the one closest to 0 is the greatest, so compare their distances from 0.

Traps

  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
  • Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Comparing negative numbers
if a<b, then −a>−b\text{if } a \lt b \text{, then } -a \gt -b
a and b are positive numbers. Putting a minus sign on both flips their order. The one with the smaller size is closer to 0, so it is greater. For example, 2 is less than 5, but −2 is greater than −5.

  1. Sort out the question

    All four numbers are negative, which means less than 0. You need the greatest one, the one with the highest value. On a number line, greater numbers are farther to the right.

  2. Think about distance from 0

    For negative numbers, the one closest to 0 is the greatest. The size of a number without its sign is its absolute value. It is the number’s distance from 0. So look for the smallest size.

    if a<b, then −a>−b\text{if } a \lt b \text{, then } -a \gt -b

  3. Compare the sizes

    The sizes are 1/49, 49, 1/7 and 1. Both 1/49 and 1/7 have 1 on top. Cutting 1 into 49 pieces gives much smaller pieces than cutting it into 7. So 1/49 is less than 1/7.

    149<17<1<49\frac{1}{49} \lt \frac{1}{7} \lt 1 \lt 49

    So 1/49 is the smallest size. That puts −149-\frac{1}{49} closest to 0.

  4. Put the numbers in order

    Flip the order of the sizes to order the negative numbers, from least to greatest.

    −49<−1<−17<−149-49 \lt -1 \lt -\frac{1}{7} \lt -\frac{1}{49}

    Watch out: −49 has the biggest digits, but it is the least, so choice B is wrong. Watch out: 1/7 is more than 1/49, but −1/7 is less than −1/49. So choice C is wrong.

    So the answer is A, −149-\frac{1}{49}.

Check your answer

Change the fractions to decimals: −149≈−0.02-\frac{1}{49} \approx -0.02 and −17≈−0.14-\frac{1}{7} \approx -0.14. Think of money you owe. Owing 2 cents is better than owing 14 cents, $1 or $49. So −1/49 is the greatest.

More questions like this

The original DOE question

Question 78 as printed on page 205 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 205, question 78. Open page 205 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).