1. Lesson #1 Math Strategies & Shortcuts
The art of choosing numbers, working backwards, and approximating to help you avoid doing tough math

2. First, this is a question from a recent actual SAT exam…did you know that approximately 88% of students will miss this question?
This is the last question in the no-calculator multiple choice section:
#15
For positive real numbers a, b, c, and d, a percent of b is the same as c percent of d. what is d in terms of a, b, and c ?
a) ab/c
b) ab/(100c)
c) ab/(1000c)
d) ab/(10000c)
wouldn’t it be great if there was more than one way to solve this highly difficult problem??
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3. SHORTCUT Definition: a method, procedure, or policy that reduces the time or energy needed to accomplish something.
For positive real numbers a, b, c, and d, a percent of b is the same as c percent of d. what is d in terms of a, b, and c ?
a) ab/c let’s reduce the time and
b) ab/(100c) energy it takes to answer
ab/(1000c) this question by transforming
ab/(10000c) it from an algebra problem into an arithmetic problem.
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4. CHOOSING NUMBERS: Most of us are better at solving problems when we are able to work with something concrete rather than something abstract…
For positive real numbers a, b, c, and d, a percent of b is the same as c percent of d. what is d in terms of a, b, and c ?
a) ab/c let’s replace each of the variables
b) ab/(100c) with a real number whose
ab/(1000c) properties (factors, multiples, etc.)
ab/(10000c) we know well and that fits the parameters in the question.
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5. For Example: Your new problem might look something like this…
For positive real numbers 20, 100, 50, and 40, 20 percent of 100 is the same as 50 percent of 40. what is 40 in terms of 20, 100, and 50 ?
a) ab/c = 2000/50 = 40 √ Once we turn this algebra problem
b) ab/(100c) = .4 into an arithmetic problem, we can
ab/(1000c) = .04 use our chosen numbers to discern
ab/(10000c) = the relationships between the variables, which is what this question is really testing.
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6. CHOOSING NUMBERS PRACTICE 1: Here are a few questions you may have struggled with from your diagnostic…
If a/b = 2, what is the value of 4b/a ?
a) 0 Once again, let’s replace the variables
b) 1 with real numbers whose
2 properties (factors, multiples, etc.)
4 we know well and that fits the parameters in the question.
Don’t forget: your TARGET is the answer to your question (i.e. what you are looking for in the answer choices) given the numbers you chose for the variables.
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7. CHOOSING NUMBERS PRACTICE 2: Shortcuts are not NECESSARILY faster; they are, however, less likely to result in a trap answer choice, which makes them more “fool proof”
If x > 3, which of the following is equivalent to
1 1 𝑥 𝑥+3 ?
a) 2𝑥+5 𝑥 2 +5𝑥 We call our tips shortcuts, because they
b) 𝑥 2 +5𝑥+6 2𝑥 will help you avoid traps, especially those
C) 2x that come from “slip ups” in performing
d) 𝑥 2 +5𝑥+6 algebraic tasks.
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8. Prowess’ s SAT Math Method: This time, when we answer the question, let’s try using a methodical approach. Great test takers have unique approaches to every question type. Our approach on the SAT math is called Q-C-T. Here’s how it works…
Q = Question (ask yourself, “What exactly is my question asking me to figure out?)
-underline or take notes
C = Concept (ask yourself, “What type of math is involved here?”
-organize the mental closet)
T = Terrain (ask yourself, “Is there a unique format to how this question is written?)
- see if you can use a shortcut should you need one
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9. The 6-Tiered Approach to achieving success on the SAT:
1 – get to know the test √
2 – learn B.T.S. (basic test-taking strategy) √
3 – learn format-specific methodology √
4 – learn the content
5 – learn advanced strategy
6 – gain consistency through practice

10. CHOOSING NUMBERS PRACTICE 3: This is another one from your diagnostic…let’s see how combining a unique approach and strategy can help us gain more points.
Alma bought a laptop computer at a store that gave a 20 percent discount off its original price. The total amount she paid to the cashier was p dollars, including an 8 percent sales tax on the discounted price. Which of the following represents the original price of the computer in terms of p ?
a) 0.88p Q =
b) p/ C =
(0.8)(1.08)p T =
p/((0.8)(1.08)) Now, we are ready to solve!
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11. CHOOSING NUMBERS PRACTICE 4: Here, you can see how using a shortcut really does help us avoid trap answers. Try this one out on your own and see if you land on the obvious choice. Then try it again. Only this time use your new shortcut.
A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollars, on 2 cars that sold for $14,000 each ?
a) 280k Q =
b) 7,000k C =
28,000k T =
(28,000 + k) / 100
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12. Great test takers always try to personalize the problem
Great test takers always try to personalize the problem. They know that by making Jane Doe’s problem their own, they will be forced to take ownership of it and, therefore, more willing to solve it. This also helps you create parallels between the problem on the page and something you might have experienced before. Let’s go ahead and personalize this problem now…
My commission is k percent of the selling price of a car. Which of the following represents the commission I will receive, in dollars, on 2 cars that sold for $14,000 each ?
a) 280k
b) 7,000k
28,000k
(28,000 + k) / 100
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13. How often do you think great test takers read over a given problem? Two? Maybe three times?
Cognitive research tells us that the human brain can only hold about 7 pieces of information at any one time. It is something called “Working Memory Capacity,” and it doesn’t matter who you are – Albert Einstein or Snooki – what you get is what you will always have. So, write down everything. Reserve that important real estate for problem processing.
Understanding just how our brains work can make us much better problem solvers on the SAT and in life. Look for our “Brain Tips” throughout there lectures, symbolized by the brain icon above.
A salesperson’s commission is k percent of the selling price of a car. Which of the following represents the commission, in dollars, on 2 cars that sold for $14,000 each ?
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14. CHOOSING NUMBERS PRACTICE 5: Try some more
CHOOSING NUMBERS PRACTICE 5: Try some more. Be sure to write down your information as you go. You might not NEED the shortcut here, but you should ALWAYS have 2 ways to solve every problem if you can. That’s what separates the good test takers from the great ones.
On Saturday afternoon, Armand sent m text messages each hour for 5 hours, and Tyrone sent p text messages each hour for 4 hours. Which of the following represents the total number of messages sent by Armand and Tyrone on Saturday afternoon?
a) 9mp Q =
b) 20mp C =
5m + 4p T =
4m + 5p don’t forget to personalize your questions!
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15. CHOOSING NUMBERS PRACTICE 6: The more you practice on your own choosing numbers whenever you see undefined variables, the more you will find unique applications. Here’s another one where, if you forget the rules or don’t know them, you might get stuck. Well, think again…this time using our shortcut.
If 3x – y = 12, what is the value of 8 𝑥 2 𝑦 ?
a) Q =
b) C =
c) T =
d) 4 2
Suppose you couldn’t use your calculator here. Could you still use choosing numbers? The answer is a resounding “Yes!” The reason involves a central strategy you will see us use time and again as we solve multiple choice problems. We call it ballparking and we will discuss it in more detail in a few minutes.
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16. CHOOSING NUMBERS PRACTICE 7: Here are more real practice test questions straight out of your blue book…NO CALCULATOR
If f(x) = - 2x + 5, what is f(-3x) equal to ?
a) - 6x Q =
b) 6x C =
c) 6x T =
d) 6 𝑥 x
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17. CHOOSING NUMBERS PRACTICE 8: NO CALCULATOR
3(2x + 1)(4x + 1)
Which of the following is equivalent to the expression above?
a) 45x Q =
b) 24 𝑥 C =
c) 𝑥 x T =
d) 18 𝑥
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18. CHOOSING NUMBERS PRACTICE 9: NO CALCULATOR
If 𝑎 −𝑏 𝑏 = , which of the following must also be true ?
a) 𝑎 𝑏 =− 4 7
Q =
b) 𝑎 𝑏 = 10 7
C =
c) 𝑎+𝑏 𝑏 = 10 7
T =
d) 𝑎 −2𝑏 𝑏 =− 11 7
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19. CHOOSING NUMBERS PRACTICE 10: NO CALCULATOR
Ken and Paul each ordered a sandwich at a restaurant. The price of Ken’s sandwich was x dollars, and the price of Paul’s sandwich was $1 more than the price of Ken’s sandwich. If Ken and Paul split the cost of the sandwiches evenly and each paid a 20% tip, which of the following expressions represents the amount, in dollars, each of them paid (assume there is no sales tax) ?
a) 0.2x Q =
b) 0.5x C =
c) 1.2x T =
d) 2.4x + 1.2
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20. CHOOSING NUMBERS PRACTICE 11: CALCULATOR OK!
If x is the average (arithmetic mean) of m and 9, y is the average of 2m and 15, and z is the average of 3m and 18, what is the average of x, y, and z in terms of m ?
a) m Q =
b) m C =
c) 2m T =
d) 3m + 21
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21. CHOOSING NUMBERS PRACTICE 12: CALCULATOR OK!
Let x and y be numbers such that –y < x < y. Which of the following must be true ?
| x | < y
X > 0
Y > 0
a) I only Q =
b) I and II only C =
c) I and III only T =
d) I, II, and III
Think smart not hard. If Roman Numeral I is in every answer choice, we obviously don’t need to test it out – it works.
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22. HOW TO SOLVE ALMOST ANYTHING: (10 PRINCIPLES)
???
> Throw things out and see what sticks
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23. CHOOSING NUMBERS PRACTICE 13: CALCULATOR OK!
Suppose there are n fish in a tank. If half of the fish are striped, s, and three quarters of the fish are red, r, in terms of s and r, which of the following cannot be a possible number of fish in the tank that are both striped and red?
| s – r |
r – s
c) 𝑟 5 + s
d) 𝑟 5 + 𝑠 5
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24. Shortcut #2: Now we have a whole new approach and a great time saving shortcut – a backup plan – to help us with those tough algebra problems. Let’s add one more shortcut, which should cover us for about a staggering 50% of all math questions.
3x + 4y = - 23
2y – x = -19
What is the solution (x, y) to the system of equations above ?
a) (-5, -2) The great thing about multiple choice is
b) (3, -8) that the test makers have already provided
(4, -6) the answer for you. Now it just becomes a
d) (9, -6) game of seek and find.
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25. WORKING BACKWARDS: This second shortcut involves using the answer choices provided to help solve the problem. In some circles, this is referred to as “plug and chug.”
Try an answer choice to see if the given numbers yield the same result as on the opposite side of the equal sign.
3x + 4y = - 23
2y – x = -19
What is the solution (x, y) to the system of equations above ?
a) (-5, -2)
b) (3, -8) notice the answer choices
(4, -6) feature numbers only. This
d) (9, -6) is our second terrain.
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26. For Example: You might keep your work next to each choice, like this…
3x + 4y = - 23
2y – x = - 19
What is the solution (x, y) to the system of equations above ?
(-5, -2) = 3(-5) + 4(-2) = = - 23
2(-2) - -5 = = 1 – nope!
(3, -8) = 3(3) + 4(-8) = = -23
= 2(-8) – 3 = - 16 – 3 = -19 – this works!
(4, -6)
d) (9, -6)
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27. WORKING BACKWARDS PRACTICE 1: Occasionally, you can minimize the amount of plugging in you have to do simply by starting in the middle. Try working backwards on this very easy problem below starting with either choice B or C and you’ll see what I mean.
If 𝑥−1 3 = k and k = 3, what is the value of x ?
a) Q =
b) C =
c) T =
d) 10
Great test takers know how to exploit the predictable nature of standardized tests. For example, most of the SAT math multiple-choice questions are arranged in numerical order, ascending or descending. Why not take advantage?
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28. WORKING BACKWARDS PRACTICE 2: On this question, we can avoid the tedious work of using substitution or combination to solve the problem. Try it both ways and see which path you prefer.
A food truck sells salads for $6.50 each and drinks for $2.00 each. The food truck’s revenue from selling a total of 209 salads and drink in one day was $ How many salads were sold that day?
a) Q =
b) C =
c) T =
d) 105
One of the biggest separating factors between great test takers and average test takers is organization. To be an elite scorer, you can’t really afford to make many mistakes, so staying organized to avoid careless errors is a must.
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29. WORKING BACKWARDS PRACTICE 3:
Katrina is a botanist studying the production of pears by two types of pear trees. She noticed that Type A trees produced 20 percent more pears than Type B trees did. Based on Katrina’s observation, if the Type A trees produced 144 pears, how many pears did the Type B trees produce?
a) Q =
b) C =
c) T =
d) 173
You probably know how to solve this question without working backwards. It is important to remember that shortcuts are not a substitute for math knowledge; they are, however, supplementary to it.
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30. WORKING BACKWARDS PRACTICE 4:
h = 𝑡 t
The equation above expresses the approximate height h, in meters, of a ball t seconds after it is launched vertically upward from the ground with an initial velocity of 25 meters per second. After approximately how many second will the ball hit the ground?
a) Q =
b) C =
c) T =
d) 5.0
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31. WORKING BACKWARDS PRACTICE 5: (NO CALCULATOR)
𝑥 𝑦 = 6
4(y + 1) = x
If (x, y) is the solution to the system of equations above, what is the value of y ?
a) Q =
b) C =
c) T =
d) 24
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32. WORKING BACKWARDS PRACTICE 6: (NO CALCULATOR)
The functions f and g, defined by f(x) = 8 𝑥 and g(x) =
-8 𝑥 , are graphed in an xy-plane. The graphs of f and g intersect at the points (k, 0) and (-k, 0). What is the value of k ?
a) 1/4 Q =
b) 1/2 C =
c) T =
d) 2
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33. WORKING BACKWARDS PRACTICE 7: (CALCULATOR OK)
Last week Raul worked 11 more hours than Angelica. If they worked a combined total of 59 hours, how many hours did Angela work last week?
a) Q =
b) C =
c) T =
d) 48
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34. WORKING BACKWARDS PRACTICE 8: (CALCULATOR OK)
Miguel has a colored egg collection. Three quarters of his eggs are green and a sixth of those green eggs are not ostrich eggs. If he has 15 green ostrich eggs, then how many eggs does he have?
a) Q =
b) C =
c) T =
d) 24
Sometimes using shortcuts exposes the necessary properties you should be looking for in an answer choice. What happened when you tried to get three quarters of 18 or 22?
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35. WORKING BACKWARDS PRACTICE 9: (CALCULATOR OK) This real test question is NOT easy. Just see for yourself. Try to solve it algebraically, first, using inequalities…
Suppose $150 dollar bills are divided up among Alice, Bob, Carl, Dave, Edward, and Fiona. If Alice receives more than any other person, then what is the least number of dollars that Alice can receive?
a) Q =
b) C =
c) T =
d) 25
Many of the highest difficulty questions can be solved using shortcuts. And in no time at all. Be sure to master these 2 great techniques.
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36. WORKING BACKWARDS PRACTICE 10: (CALCULATOR OK)
𝑥 𝑦 2 < 6
x + y > 4
If x and y are positive integers in the inequalities above and x > y, what is the value of x ?
a) Q =
b) C =
c) T =
d) 4
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37. WORKING BACKWARDS PRACTICE 11: (CALCULATOR OK)
Let f(x) be defined as f(x) = 𝑥 2 - x for all values of x.
If f(b) = f(b - 2), what is the value of b ?
a) Q =
b) 1/2 C =
c) 3/2 T =
d) 3
hint: try converting from fractions to decimals. Always know which one will be easier for you to work with when presented with one or the other.
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38. WORKING BACKWARDS PRACTICE 12: (CALCULATOR OK)
n(t) = 𝑡 t + k
There was a 100-day period when the number of bees in a certain hive could be modeled by the function n above. In the function, k is a constant and n(t) represents the number of bees on day number t for 0 ≤𝑡≤ 99. On what number day was the number of bees in the hive the same as it was on day number 10 ?
a) Q =
b) C =
c) T =
d) 50
You may be asking, “Ok, but can I use these 2 shortcuts on geometry or trig problems?” The answer is YES but, on the new SAT, these problem types are now so rare that they only make up a combined 8% of the math test. The new SAT is all about algebra and stats.
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39. BALLPARKING: Remember how way, way back we said that we would revisit this question and the strategy we call ballparking? Well, here we are!
If 3x – y = 12, what is the value of 8 𝑥 2 𝑦 ? a)
b) 4 4 c)
d) 4 2
But first…
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40. PUT AWAY YOUR BOOKS—IT IS TIME FOR THE… P.O.T.D!!!
(Puzzle of the Day)
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41. How many golf balls would it take to fill a standard-sized school bus?
POTD #1:
How many golf balls would it take to fill a standard-sized school bus?
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42. That you are given the ability to approximate is the true beauty of multiple choice
Approximately, how many golf balls would it take to fill a standard-sized school bus?
a) 5,363 Bonus: what is this??
b) 53,633
c) 536,338
d) 5,363,381
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43. HOW TO SOLVE ALMOST ANYTHING: (10 PRINCIPLES)
???
Throw things out and see what sticks
Use known values to procure a reasonable range
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44. Each wrong answer choice is off by a magnitude of 10
Each wrong answer choice is off by a magnitude of 10. That’s a long way off!
Approximately, how many golf balls would it take to fill a standard-sized school bus?
a) 5,363
b) 53,633
c) 536,338
d) 5,363,381
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45. BALLPARKING: If you want to be among the best at taking the math portion of the SAT, ballparking is a skill you need to master and employ regularly!
If 3x – y = 12, what is the value of 8 𝑥 2 𝑦 ? a)
b) 4 4 c)
d) 4 2
Notice how we didn’t need to get to an EXACT value to answer this question. Approximating can save us valuable time which might otherwise be spent doing tedious calculations, especially when we can’t use a calculator. With 58 questions, every 1 second saved per question = 1 minute saved per test.
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46. BALLPARKING PRACTICE 1: Nothing is more frustrating than doing a minute of work and then finding your answer doesn’t match anything in the choices.
If 68.5x = 493.2, what is the value of x ?
a) 8.6
b) 7.2
c) 7
d) 6.1
Hint: If this question read 70x = 490, what would x be then?
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47. The 6-Tiered Approach to achieving success on the SAT:
1 – get to know the test √
2 – learn B.T.S. (basic test-taking strategy) √
3 – learn format-specific methodology √
4 – learn the content
5 – learn advanced strategy √
6 – gain consistency through practice
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48. BALLPARKING PRACTICE 2: Sometimes we can ballpark obvious angles or determine the length of a side in a figure drawn to scale just by looking.
s t l m
In the figure above, lines l and m are parallel and lines s and t are parallel. If the measure of angle 1 is 35◦, what is the measure of angle 2 (in degrees) ?
a) 35
b) 55
c) hint: you don’t need to subtract 35 from 180
d) 145
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49. BALLPARKING PRACTICE 3: Make sure you know certain percentages by heart... 1% and 10% (NO CALCULATOR)
Nick surveyed a random sample of the freshman class of his high school to determine whether the Fall Festival should be held in October or November. Of the 90 students surveyed, 25.6% preferred October. Based on this information, about how many students in the entire 225-person class would be expected to prefer having the Fall Festival in October?
a) 50
b) 60
c) 75 80
A Great test taker might solve this question before he even finished reading it. An average test taker will take over a minute and still might not solve it correctly.
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50. The 6-Tiered Approach to achieving success on the SAT:
1 – get to know the test √
2 – learn B.T.S. (basic test-taking strategy) √
3 – learn format-specific methodology √
4 – learn the content
5 – learn advanced strategy √
6 – gain consistency through practice √
Lesson 1

51. LET’S SUMMARIZE: Lesson 1
It is a fallacy to think that you have to answer SAT math questions using the standard approaches you’ve learned in high school. You’ll actually find that, unless you want to be a mathematician or an engineer, you will very rarely use such methods once you graduate.
Choosing numbers is a shortcut that involves using concrete values rather than abstract variables in order to solve problems. Great test takers know this is often faster and always more “fool proof” than grinding it out with algebra.
Working backwards is a shortcut that takes advantage of the fact that the SAT is a multiple choice test. Great test takers know that it makes more sense to use the narrowed-down field of possible answers that the test makers give them than to figure the problem out from scratch.
Ball parking is not a shortcut but a kind of necessary life skill. If you were building a porch or cutting something to fit perfectly into a given space, you would need to by as accurate as possible in your measurement. But, in most scenarios, just having a broad idea is enough. On multiple choice, this allows you to narrow the scope down to only one or two reasonable answer choices.
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