Every year, 7^{th} graders from throughout New York take the Common Core state math exam. While students will often have taken a similarly formatted exam every year since 3^{rd} grade, the 7^{th} grade state exam has by far the highest stakes. While the specialized high schools may exclusively look at the Specialized High School Admission Test(SHSAT), many of the other schools that students will be applying to focus heavily on the state exam. Among the NYC public schools that look at state exam scores as a critical function of their admissions regime are Beacon, Bard, Townsend Harris, and nearly all other schools seen on the top of the admission application ticket. For nearly all these schools, the average admitted student will have gotten a 3.5 or higher score, which means getting even the hardest questions on the exam correct is critical!

After each exam year, the state releases a truncated version of the official exam along with the percentage of students who got each question correct. For all of the questions blow, fewer than 60% of the students taking the exam got the questions correct, but with some study and preparation, all ten can be easily mastered.

**#10 with 35% of students answering correctly: 2016, Question 39**

Line KN represents a proportional relationship. Point *N *lies at **(18, 12)**, as shown on the graph below.

Which ordered pair could represent the coordinates of point K?

**A. (6, 0)**

**B. (2, 3)**

**C. (1.5, 0)**

**D. (7.5, 5)**

**Click to see the answer and explanation!**

(D) If you answered “D”, then you got this difficult 2016 question correct! As is the case with many of the questions on this list, the 10^{th} hardest question of the last five years tests multiple mathematical concepts simultaneously. In order to get this question correct, you must know 1) the definition of a proportional relationship; 2) Know the relationship between x and y on an ordered pair; and 3) Be capable of managing proportions that have partial numbers.

1: A proportional relationship is one where the quotient of x divided by y is always the same. In order for this to be the case, the y-intercept—the point at which a line or parabola intersects with the y-axis—must be 0.

2: The relationship between x and y on a linear pair will always be (X, Y). In this question, students should be able to remind themselves of this relationship by seeing that the 18 in the question coincides with the 18 on the x-axis. Reminder: The question part of a question will ALWAYS be correct, so can be used as a tool to remind oneself of mathematical truisms.

3: In this question, the relationship between x and y is 1.5. Answer D is the only question that has this relationship.

**#9 with 35% of students answering correctly: 2017, Question 36**

Jeanette purchased a concert ticket on a web site. The original price of the ticket was $75. She used a coupon code to receive a 20% discount. The web site applied a 10% service fee to the discounted price. Jeanette’s ticket was less than the original price by what percent?

A. 7%

B. 10%

C. 12%

D. 28%

**Click to see the answer and explanation!**

(C) Every year, the seventh grade state test contains a multitude of proportion questions. Often, these questions will be among the problem types that teachers focus on the most heavily during the course of the academic year. However, while students will study various proportion questions endlessly, teachers will often teach them in a fairly rigid manner that may lead to a greater proportion of students failing to land on the right answer when the question is slightly altered from the most common versions of the question.

In this instance, students must first remove 20% from the original price, then add 10% of the new value to the price, and then find the ratio of the amount saved to the original. To do so, a student can either use proportions or algebraic equations. To use proportions, a student would first establish a proportion of 80/100 = x/75 to find that x = 60. Doing so would show that a 20% reduction in the original price would lead to an item that costs $60. Next, the student would establish a proportion of 110/100 = x/60 to find that x = 66. Here, they are learning that when you add 10% to the new price, you end at $66. For most students, the difficulty in setting up the proportion will be in realizing that you are trying to find 20% *less *than the original and then 10% *more *than that amount. Once you determine that the price of the new product will be $66, students should use the proportion 9/75 = x/100 to learn that the original price was reduced by 12% by the end. In this proportion, 9 was determined by subtracting 66—the final price—from 75—the original price.

For students who can convert the percentages into decimals for multiplication purposes, the equation would be (75-(75 * .8 * 1.1))/75. In this instance, the value is being reduced by 20%, then increased by 10%. Then that new number is being subtracted from the original. Finally, the new value is being divided by the original so that a percent change can be determined. Most students will want to try doing this instead of the proportions, but they should keep in mind that one of the main benefits of using proportions is to create opportunities for falsifying your work, saving you from making a wrong computation. As always, we suggest using proportions whenever possible so as to limit the failure rate.

**#8 with 35% of students answering correctly: 2018, Question 23**

A farm grew 19.8 tons of wheat in 2013. The farm’s wheat output increased by 9.8% from 2013 to 2014 and by 5.1% from 2014 to 2015. Which expression represents a strategy for estimating the total output of wheat, in tons, in 2015?

A. 20 + 10 + 5

B. 20(10)(5)

C. 20 + 1.1 + 1.05

D. 20(1.1)(1.05)

**Click to see the answer and explanation!**

(D) With only 35% of students who took the exam landing on the correct answer, it is safe to say that students struggle converting problems normally solved using a proportion into expressions that can be used algebraically. For those ultimately studying to master the SHSAT, we recommend that you be able to easily convert mathematical expression into many forms. In this instance. Student must be aware that in order to increase a value by 10%, they can multiply it by 1.1, the equivalent of 110%. Most students will try finding 10% more by multiplying a figure by .1 and then adding it to the original, but this is not the most direct mathematical process, and a failure to be dept at these conversions in the simplest form will lead to a wrong answer here. After increasing the output by 10% in the first year, wheat was then produced 5% more the next year. To increase a figure by 5%, one can multiply the value by 1.05. The full equation should thus be 20 * 1.098 * 1.051. To estimate, you can round these figures to 20 * 1.1 * 1.05.

**#7 with 34% of students answering correctly: 2017, Question 34**

A bowling team participates in a two-day tournament and records the scores for each team member on both days. The scores for both days are represented by the box plots below.

Which conclusion can be drawn from the box plots?

A. The scores on Friday and the scores on Saturday have the same median and interquartile range.

B. The scores on Friday have a greater median and a greater interquartile range than the scores on Saturday.

C. The scores on Friday have a greater interquartile range than the scores on Saturday, but both data sets have the same median.

D. The scores on Friday have a greater median than the scores on Saturday, but both data sets have the same interquartile range.

**Click to see the answer and explanation!**

(D) In 2017, students may have struggled more on this box and whisker plot graph than other years, but almost all students are guaranteed to see one question that tests their knowledge of these data visualization tools. In order to answer this question correctly, students must first know how box and whisker charts are constructed. A box and whisker chart is a data visualization tool that takes a data set and visualizes the quadrants and median through use of five lines and a box. The lowest line is the smallest number in a data set. The next line shows the median of the bottom half of the values in the data set. The third line shows the median of the data set. The fourth line indicates the median of the upper half of values. Finally, the fifth number demonstrates the largest, or maximum, value in the data set. The box, encompassing the second and third quadrants, demonstrates the interquartile range.

For this question, students must first recognize that the median on Friday—215—is larger than Saturday—205. Next, they must recognize that despite the interquartile range being larger on Friday, the actual range of both is the same at 20.

**#6 with 33% of students answering correctly: 2016, Question 14**

A company ordered 10 boxed lunches from a deli for $74.50. Each boxed lunch cost the same amount. Which equation represents the proportional relationship between *y, *the total cost of the boxed lunches, and *x, *the umber of boxed lunches?

A. 7.45*x* = *y*

B. 7.45/*x* = 10/*y*

C. 74.50*x* = *y*

D. 74.50/*x* = 10/*y*

**Click to see the answer and explanation!**

(A) Questions that test students’ ability to think algebraically almost always have among the lowest percentage of students answering correctly. Nonetheless, for students ultimately aiming for the SHSAT, this question type is among the most similar to those found on the later exam. In this instance, students must know that a proportional relationship is one in which for every instance of x, there is a constant rate of increase. In this instance, the company ordered 10 boxes(x) for 74.50. To show this value, one can either show the fraction 74.5/10, or simplify to 7.45. For this problem, for every x, the company must pay 7.45 y.

**#5 with 33% of students answering correctly: 2017, Question 16**

If the expression below has a positive value, which inequality represents all possible values of *x *in the expression?

-3x

A. *x < 0*

B. *x > 0*

C. *x ≤ 0*

D. *x *≥ 0

**Click to see the answer and explanation!**

(A) Students who have already taken a SHSAT practice exam may be accustomed to “quantitative knowledge” questions such as these. For this question, students need to know that in all instances, a positive integer is generated when two negatives are multiplied. As -3 is a constant in this expression and the question states that all values are positive, x must always be negative.

If you run into a quantitative knowledge question that you are unsure of, they are often great candidates for guessing and checking or plugging in. In this case, try values from each inequality. Will any value greater than 0 times -3 give a negative number? If -3 is multiplied by 0, a value possible in answers C and D, will the answer of 0 be positive? On the state exam where students will always be in possession of a calculator, they shouldn’t be afraid of trying out values in the hopes of discovering a rule they may have forgotten or never learned.

**#4 with 32% of students answering correctly: 2015, Question 54**

A pile of newspapers in Ms. McGrath’s art class was 17 ^{3}/_{4} inches high. Each consecutive week, for the next 5 weeks, the height of the pile of newspapers increased by 8 ^{7}/_{12} inches. What was the height, in inches, of the pile after 3 week?

A. 25 ^{3}/_{4}

B. 26 ^{1}/_{4}

C. 42 ^{1}/_{4}

D. 43 ^{1}/_{2}

**Click to see the answer and explanation!**

(D) While a basic arithmetic question on its face, this question has many opportunities for students to get tripped up. However, with the knowledge of multiplying fractions, every student can get similar questions correct come test day. First, while it may be tempting to rely on your calculator, you should first try to solve this on paper. Paper has the benefit of forcing you to see the steps and catch any miscomputation. In this instance, converting the fractions into partial numbers will also give a repeating decimal. Instead, multiply the fraction ^{7}/_{12} by 3, getting ^{21}/_{12} and 8 by 3, getting 24. Now, to add 24 ^{21}/_{12} to 17 ^{3}/_{4}, you can either convert all the values into improper fractions with a common denominator of 12, or add 17 to 24, to get 41 and then convert ^{3}/_{4} to ^{9}/_{12} and add that value to ^{21}/_{12} to get ^{30}/_{12}. If you add ^{30}/_{12} to 41 and simplify you will find the answer, D.

**#3 with 31% of students answering correctly: 2015, Question 18**

During a sale, a store offered a 40% discount on a particular camera that was originally priced at $450. After the same, the discounted price of the camera was increased by 40%. What was the price of the camera after this increase?

A. $252

B. $360

C. $378

D. $450

**Click to see the answer and explanation!**

(C) No, your *de ja vu *is not misplaced. This question is incredibly similar to the 9th most difficult question above! Yet again, we see a percent change question that, due to having slight variances from the standard approaches taught in school, most students fail to answer correctly. Since this question is so similar to questions seen across the exam, and since the mistake that leads to answers A and D are so frequently duplicated on the SHSAT and common core state test exam, let’s investigate those answer choices in much the same way we do on our SHSAT School problems.

A) Answer A is achieved when a student finds 40% of 450 and then adds 40% to that amount. This is commonly done due to learning that 40% is equal to .4. However, if you are *removing *40%, then you need to multiply the value by 100-40, or 60%.

D) If you answered D, then you are not alone! When we visit schools across NYC, students often answer with quantities that replicate the mistake that leads to 40. In order to get 450, one must remove 40% from 450, and then add 40% of 450 back to that amount. However, when you do a percent change to a value, the new percent change will be of a different value.

In this case, you need to multiple 450 * .6 and then multiply that value by 1.4 to find the correct answer.

**#2 with 30% of students answering correctly: 2017, Question 32**

The scale drawing of a field in the shape of a triangle is shown below.

**Click to see the answer and explanation!**

(C) Do you know the formula for area of a triangle? If you do, you likely would have been in the 30% of students who got this answer correct. The formula for the area of a triangle is Area = (Base * Height) / 2. In this case, the base extends from point (2, 2) to (9, 2), for a total of 7. To find the height of a triangle, you should always look for the point furthest from base. In this case, the point (4, 7) is the highest point on the triangle. As the base was located on the y axis of 2, we can determine that the height is 5.

If the base is 7 and the height is 5, then we can substitute these values into the equation to find that (5 * 7) / 2 = 17.5. However, with this we’re not quite done! Looking at the key, we see that for each unit of measurement on the graph, the scale is 2 meters. As such, the answer is 17.5 * 2 = 35.

**#1 with 29% of students answering correctly: 2017, Question 21**

Which expression is equivalent to the expression shown below?

– ^{1}/_{2 }[-^{3}/_{2}*x* + 6*x* + 1] – 3*x*

A. ^{3}/_{2}*x* – ^{1}/_{2}

B. 6 ^{3}/_{4}*x* – ^{1}/_{2}

C. –^{3}/_{4}*x* + ^{1}/_{2}

D. -5 ^{1}/_{4}*x* – ^{1}/_{2}

**Click to see the answer and explanation!**

Here it is, the question that had the most students scratching their head over the last five years. Before we dive into the question, a reminder is in order for questions like this: On the state exam you have a calculator! As such, if you are ever unsure if you are remembering your order of operations correctly or if there is any other reason why you may think that you are about to answer incorrectly, you can always substitute in a value for x in the expression. If you use the same value for x in the answer choices, then it **must **give the same value if it is an equivalent expression. On the SHSAT, this strategy can also be employed in times of absolute need, but without a calculator it will have more computations and thus more possibilities for miscalculations.

For this question, first convert -3/2 to -1.5. Next, simplify within the brackets. -1.5x + 6x = 4.5x. As 1 is a constant value it cannot combine with x, so we are left with -.5(4.5x + 1) – 3x. Next, distribute -.5 to both 4.5x and 1. -.5 * 4.5x = -2.25x and -.5 * 1 = -.5. At this point, we have -2.25x – .5 – 3x. Combine like terms to get the answer, -5.25x – .5.

How many were you able to get correct? If you have questions about the 7th grade state math exam or anything else related to high school admissions, then you can message the author at jessegm@shsatschool.com. Interested in tutoring to ensure you have mastered all the material? Check out our Tutoring page! Finally, if you are ready to enroll in SHSAT School and have access to an unlimited amount of questions then you can Take a Free SHSAT Practice Test and get started!