SAT Math Hard Questions and
Answers 2026-2027
Quick Quiz 1 #5
If (c-d)/c = 5/8, which of the following must also be true?
A) (c-d)/d = 8/5
B) d/c = 13/8
C) (c+d)/c = 13/8
D) c/d = 8/3 -CORRECT ANSWER ->>C Plugging in is all about making your life easier, so do the easiest
thing here and make c = 8. Now figure out what value of d makes the equation true. The numerators
must be equal, so 8 -CORRECT ANSWER ->>d = 5 and d = 3. Use these values to test the answer choices.
Choice (A) becomes (8-5)/5 = 8/5, which is not true. Choice (B) becomes 5/8 = 13/8, which can be
eliminated. Choice (C) becomes (8+5)/8 = 13/8, which is true, and (D) becomes 8/5 = 8/3 , which is false.
Only (C) is true, so that's the correct answer.
Quick Quiz 1 #29
2x + p = 7x -CORRECT ANSWER ->>3
2y + q = 7y -CORRECT ANSWER ->>3
In the above equations, p and q are constants. If q is 5 less than p, which of the following statements is
true?
A) x is 1 less than y
B) x and y are equal
C) x is 1 more than y
D) x is 2 more than y -CORRECT ANSWER ->>C There is a lot more going on in this question, so write
things down to avoid confusion. Start with q and p since their relationship is given. If p = 7, then q = 2.
Plug these values into the equations and solve for x and y. The first equation becomes 2x + 7 = 7x -
CORRECT ANSWER ->>3 or 5x = 10, so x = 2. The second equation becomes 2y + 2 = 7y -CORRECT
ANSWER ->>3 or 5y = 5, so y = 1. Use POE to get rid of answers that aren't true. All but (C) are false, so
(C) is the correct answer.
Quick Quiz 2 #24
,In the figure shown above, sin (c°) = cos (d°). If c = 3y -CORRECT ANSWER ->>27 and d = 7y -CORRECT
ANSWER ->>16, what is the value of y ?
A) 3.3
B) 5.3
C) 10.3
D) 13.3 -CORRECT ANSWER ->>D This is a hard one that expects you to know some things about
trigonometry. You may not recognize the rule being tested, but you should see the opportunity to use
PITA. The answers represent y, so plug them into the equations for c and d and get those values. Then
it's just a matter of using your calculator's handy SIN and COS buttons to see if sin (c°) = cos (d°). Start
with (C): if y = 10.3, c = 3(10.3) -CORRECT ANSWER ->>27 = 3.9 and d = 7(10.3) -CORRECT ANSWER ->>16
= 56.1. Use your calculator, in degree mode, to find that sin (3.9°) = 0.068 and cos (65.1°) = 0.421. These
aren't equal, so eliminate (C). It may be hard to tell which way to go, but at worst, you'll have to try
three of the four answers, which should still be worth the time. In this case, a smaller number will make
c negative, which probably won't work, so go bigger. Now try (D): if y = 13.3, c = 12.9 and d = 77.1. Since
sin (12.9°) = 0.223 and cos (77.1°) = 0.223, (D) is correct. You may notice how 12.9° and 77.1° add up to
90°. This question is really testing if you know that for the complementary angles in a right triangle, the
sine of one is equal to the cosine of the other, and vice versa. Using PITA, though, you could avoid that
entirely.
Quick Quiz 3 #21
Three numbers, a, b, and c, have a sum of 672. The value of a is 25% less than the sum of b and c. What
is the value of a?
A) 97
B) 135
C) 288
D) 372 -CORRECT ANSWER ->>C Start by estimating. The value of a can't be too small, if it is close to the
sum of b and c. Choice (A) is probably too small, so eliminate it. The value of a also can't be more than
half the total, since it is less than the sum of the other two numbers. Therefore, (D) is too big. With only
(B) and (C) left, you could just guess and go, but using PITA will get you the right answer. For (C), if a =
288, there is 672 -CORRECT ANSWER ->>288 = 384 left for the sum of b and c. Is 288 = 384 -CORRECT
ANSWER ->>25/100 (384)? Yes, it is, so (C) is the correct answer.
Quick Quiz 1 Ch. 2 #23
An art teacher has a jar containing b buttons to distribute to her class for a project. In order to give each
child in the class 6 buttons, she will need 18 more buttons. If she gives each child 5 buttons, she will
have 4 left over. How many children are in the class?
, A) 17
B) 19
C) 22
D) 24 -CORRECT ANSWER ->>C Plug In the Answers, starting with one of the middle ones. If there are 22
students in the class, the teacher will need 22 × 6 = 132 buttons in order to give each child 6 buttons.
The question says she is 18 buttons short, so subtract 18 from 132 to find that b = 114 buttons. In the
second scenario, she has 4 buttons left over, so subtract 4 to get 110. Is that divisible by 5 to give 5
buttons to each child? Yes! 110 divided by 5 is 22, the number of children in the class. So (C) works and
is the correct answer.
Quick Quiz 2 Ch. 2 #28
At a track meet, 2/5 of the first-place finishers attended Southport High School, and of 1/2 them were
girls. If 2/9 of the first-place finishers who did NOT attend Southport High School were girls, what
fractional part of the total number of first-place finishers were boys?
A) 1/9
B) 2/15
C) 3/5
D) 2/3 -CORRECT ANSWER ->>D Plug In. The total number of first-place finishers was 30. You can find
the number who were from Southport by taking 2/5 of 30 = 12. That leaves 18 who did not go to
Southport High School. If half the 12 Southport runners were girls, that means 6 were girls and 6 were
boys. If 2/9 of the non-Southport runners were girls, then 2/9 of 18 = 4 girls, which leaves 14 boys. That
means a total of 14 + 6 = 20 boys, out of a total of 30, or 20/30 = 2/3.
Quick Quiz 3 Ch.2 #1
If 0.2p = 4, what is the value of 4p ?
A) 2
B) 8
C) 40
D) 80 -CORRECT ANSWER ->>D If 0.2p = 4, then divide both sides by 0.2 to get p = 20. Then, multiply
both sides by 4 to get 4p = 80.
Quick Quiz 3 Ch.2 #29
For all values of a, b, c, and d, which of the following is equivalent to (ad/bc)/(ac/bd)?
Answers 2026-2027
Quick Quiz 1 #5
If (c-d)/c = 5/8, which of the following must also be true?
A) (c-d)/d = 8/5
B) d/c = 13/8
C) (c+d)/c = 13/8
D) c/d = 8/3 -CORRECT ANSWER ->>C Plugging in is all about making your life easier, so do the easiest
thing here and make c = 8. Now figure out what value of d makes the equation true. The numerators
must be equal, so 8 -CORRECT ANSWER ->>d = 5 and d = 3. Use these values to test the answer choices.
Choice (A) becomes (8-5)/5 = 8/5, which is not true. Choice (B) becomes 5/8 = 13/8, which can be
eliminated. Choice (C) becomes (8+5)/8 = 13/8, which is true, and (D) becomes 8/5 = 8/3 , which is false.
Only (C) is true, so that's the correct answer.
Quick Quiz 1 #29
2x + p = 7x -CORRECT ANSWER ->>3
2y + q = 7y -CORRECT ANSWER ->>3
In the above equations, p and q are constants. If q is 5 less than p, which of the following statements is
true?
A) x is 1 less than y
B) x and y are equal
C) x is 1 more than y
D) x is 2 more than y -CORRECT ANSWER ->>C There is a lot more going on in this question, so write
things down to avoid confusion. Start with q and p since their relationship is given. If p = 7, then q = 2.
Plug these values into the equations and solve for x and y. The first equation becomes 2x + 7 = 7x -
CORRECT ANSWER ->>3 or 5x = 10, so x = 2. The second equation becomes 2y + 2 = 7y -CORRECT
ANSWER ->>3 or 5y = 5, so y = 1. Use POE to get rid of answers that aren't true. All but (C) are false, so
(C) is the correct answer.
Quick Quiz 2 #24
,In the figure shown above, sin (c°) = cos (d°). If c = 3y -CORRECT ANSWER ->>27 and d = 7y -CORRECT
ANSWER ->>16, what is the value of y ?
A) 3.3
B) 5.3
C) 10.3
D) 13.3 -CORRECT ANSWER ->>D This is a hard one that expects you to know some things about
trigonometry. You may not recognize the rule being tested, but you should see the opportunity to use
PITA. The answers represent y, so plug them into the equations for c and d and get those values. Then
it's just a matter of using your calculator's handy SIN and COS buttons to see if sin (c°) = cos (d°). Start
with (C): if y = 10.3, c = 3(10.3) -CORRECT ANSWER ->>27 = 3.9 and d = 7(10.3) -CORRECT ANSWER ->>16
= 56.1. Use your calculator, in degree mode, to find that sin (3.9°) = 0.068 and cos (65.1°) = 0.421. These
aren't equal, so eliminate (C). It may be hard to tell which way to go, but at worst, you'll have to try
three of the four answers, which should still be worth the time. In this case, a smaller number will make
c negative, which probably won't work, so go bigger. Now try (D): if y = 13.3, c = 12.9 and d = 77.1. Since
sin (12.9°) = 0.223 and cos (77.1°) = 0.223, (D) is correct. You may notice how 12.9° and 77.1° add up to
90°. This question is really testing if you know that for the complementary angles in a right triangle, the
sine of one is equal to the cosine of the other, and vice versa. Using PITA, though, you could avoid that
entirely.
Quick Quiz 3 #21
Three numbers, a, b, and c, have a sum of 672. The value of a is 25% less than the sum of b and c. What
is the value of a?
A) 97
B) 135
C) 288
D) 372 -CORRECT ANSWER ->>C Start by estimating. The value of a can't be too small, if it is close to the
sum of b and c. Choice (A) is probably too small, so eliminate it. The value of a also can't be more than
half the total, since it is less than the sum of the other two numbers. Therefore, (D) is too big. With only
(B) and (C) left, you could just guess and go, but using PITA will get you the right answer. For (C), if a =
288, there is 672 -CORRECT ANSWER ->>288 = 384 left for the sum of b and c. Is 288 = 384 -CORRECT
ANSWER ->>25/100 (384)? Yes, it is, so (C) is the correct answer.
Quick Quiz 1 Ch. 2 #23
An art teacher has a jar containing b buttons to distribute to her class for a project. In order to give each
child in the class 6 buttons, she will need 18 more buttons. If she gives each child 5 buttons, she will
have 4 left over. How many children are in the class?
, A) 17
B) 19
C) 22
D) 24 -CORRECT ANSWER ->>C Plug In the Answers, starting with one of the middle ones. If there are 22
students in the class, the teacher will need 22 × 6 = 132 buttons in order to give each child 6 buttons.
The question says she is 18 buttons short, so subtract 18 from 132 to find that b = 114 buttons. In the
second scenario, she has 4 buttons left over, so subtract 4 to get 110. Is that divisible by 5 to give 5
buttons to each child? Yes! 110 divided by 5 is 22, the number of children in the class. So (C) works and
is the correct answer.
Quick Quiz 2 Ch. 2 #28
At a track meet, 2/5 of the first-place finishers attended Southport High School, and of 1/2 them were
girls. If 2/9 of the first-place finishers who did NOT attend Southport High School were girls, what
fractional part of the total number of first-place finishers were boys?
A) 1/9
B) 2/15
C) 3/5
D) 2/3 -CORRECT ANSWER ->>D Plug In. The total number of first-place finishers was 30. You can find
the number who were from Southport by taking 2/5 of 30 = 12. That leaves 18 who did not go to
Southport High School. If half the 12 Southport runners were girls, that means 6 were girls and 6 were
boys. If 2/9 of the non-Southport runners were girls, then 2/9 of 18 = 4 girls, which leaves 14 boys. That
means a total of 14 + 6 = 20 boys, out of a total of 30, or 20/30 = 2/3.
Quick Quiz 3 Ch.2 #1
If 0.2p = 4, what is the value of 4p ?
A) 2
B) 8
C) 40
D) 80 -CORRECT ANSWER ->>D If 0.2p = 4, then divide both sides by 0.2 to get p = 20. Then, multiply
both sides by 4 to get 4p = 80.
Quick Quiz 3 Ch.2 #29
For all values of a, b, c, and d, which of the following is equivalent to (ad/bc)/(ac/bd)?
