Tip #45 to Get a Top ACT Math Score

"Careless errors are bad mmmkay." So, let's learn to avoid five more of the most common ones.

(x + 3)² = (x + 3)(x + 3) = x² + 6x + 9, not x² + 9.

Remember the middle term when you FOIL.

2(3)² = 18, not 36.

Remember your old friend PEMDAS, order of operation.

Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction

Remember to finish the question. Sometimes you have gotten a number for part of a question and the ACT provides that number as an answer choice. So before you grid each answer, take a moment to pause and say, "Wait (insert your name here), did I finish the question?" This is so simple, but hugely important. This is the most common careless error, and it's so easy to avoid!
(x²)(4x⁵)(3x) = 12x⁸ not 12x⁷.

Remember that x without an exponent means x¹.

Lots of ACT questions mention units, like feet or centimeters. And 90% of these are just there to make the question realistic and are not at all tricky. But when a question has feet and inches or hours and minutes in the same question, remember to convert correctly. For example, 3.5 feet is not 3 feet 5 inches; it is 3 feet 6 inches. And 2.5 hours is not 2 hours 50 minutes; it is 2 hours 30 minutes. Of course you know this, but it's easy to forget when rushing to solve a question. As long as you know to keep an eye out for it, you'll spot it when it comes up.

Students lose way too many points on careless errors. Don't do that. Memorize the above rules and practice watching for them. Be focused, relaxed, and mindful, and you will not make careless errors. You will be a math ninja, and like most ninja of ancient Japan, you will ace the ACT.

Let's look at this question:

Solution: Great question. So easy to make a careless error, unless you know to watch for it. To divide 9 feet 10 inches by 2, let's do feet and inches separately. Here's the key, 9 feet divided by 2 is 4'6", not 4'5", and 10 inches divided by 2 is 5". So 4'6" plus 5" equals 4'11". Nasty tricksy Hobbitses.

Correct answer: B

Example Problems

Medium

If f(x) = 3x², which of the following expresses f(4p)?
1. 12p²
2. 32p²
3. 48p²
4. 64p²
5. 144p²
In a timed run, Maiko posted a time of 1 minute 45 seconds for a quarter-mile course. If she could maintain that pace, how many minutes would it take her to run 1 mile?
1. 8.2
2. 7.0
3. 6.4
1. 5.8
2. 2.9

Hard

If x = 3y, then which of the following is equivalent to (x – 2)²?
1. 3y – 2
2. 3y² – 4
3. 3y² + 6y – 4
4. 9y² – 12y + 4
5. 9y² + 18y + 4
For functions p and r such that p(x) = 2x(x – 2) and r(x) = , find r(p(–4)).
1. 0.5
2. 10
3. 12
1. 14
2. 16
Josh rode his bicycle from his house to his grandmother's house at a rate of 10 miles per hour. Later that day he rode home at a rate of 15 miles per hour. If the round trip took a total of 30 minutes, how many miles does Josh live from his grandmother?
1. 0.3
2. 3.0
3. 4.0
1. 4.5
2. 8

Answers

C f(4p) means plug 4p in for x. So f(4p) = 3(4p)² = 3(16p²) = 48p². Careless Error Buster: Remember that the 4 gets squared before you multiply by 3. Order of operations, my man.
G Since she ran a quarter (one-fourth) of a mile in 1 minute 45 seconds, to determine her time for a full mile, we multiply by 4. And 1 minute 45 seconds times 4 is not (1.45)(4) = 5.8, it is (1.75)(4) = 7, since 45 minutes is three-quarters (0.75) of an hour.
D Plug in 3y for x in the expression (x – 2)² to get (3y – 2)² = 9y² – 12y + 4. Careless Error Buster: When you FOIL a binomial, remember the middle term.
K Plug –4 into the p function: p(–4) = 2(–4)(–4 – 2) = 48. Then plug 48 into the r function: r(48) = =16. Careless Error Buster: Follow the order of operations.
B Most kids have a tough time with this one. The key is that the distance is equal for both trips, so we set the (rate)(time) equal for the way there and the way back:

(Rate)(time) = (rate)(time)

Since the total trip was 0.5 hour, if we call one way t, then the return trip we can call (0.5 – t).

Notice that we should use hours, not minutes, since rate is in miles per hour.

10(t) = (15)(0. 5 – t)

10t = 7.5 – 15t

25t = 7.5

t = 0.3 hour

Now remember, the question asks for how far Josh lives, NOT the time, so D = RT and D = 10 mph (0.3 hour) = 3 miles. Careless Error Buster: Finish the question, don't stop with t = 0.3. Remember to ask, "Did I finish the question?"

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