SAT Problem-Solving & Data Analysis: method and traps

The arithmetic in this area is the easiest on the test. The difficulty is entirely in deciding what to compute. A question will give you five numbers and ask for a relationship between two of them, and the wrong pair is always available as a choice.

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This area at a glance

6 worked examples · 6 traps

112

questions

  • Foundation 21
  • Standard 66
  • Challenge 25

What this area asks for

  • Unit rates and constant-rate problems
  • Percent increase, decrease, and reversal
  • Probability from counts
  • Mean, median, and range
  • Reading two-way and one-way frequency data
  • Proportional reasoning and scale

The method

One routine that applies across the whole area, rather than a rule for each question type.

  1. 1

    Write the fraction before you divide

    Nearly every question here is one fraction. Deciding what goes on top and what goes underneath is the whole task; once written, the arithmetic is trivial. "What fraction of bus riders walk" puts bus riders underneath, not all students.

  2. 2

    Anchor a percent to the amount it applies to

    A percent is meaningless without its base. A 10 percent rise followed by a 10 percent fall does not return to the start, because the fall is taken from the larger amount. Write down which number each percent is a percent of.

  3. 3

    Reverse a percent by dividing, not subtracting

    If a price rose 25 percent to reach 80, the original is 80 divided by 1.25, not 80 minus 25 percent of 80. The second calculation is a different question, and its answer is among the choices.

  4. 4

    Use the sum when a mean changes

    The mean is the sum divided by the count, so any question about a changing mean is really about a changing sum. Convert to totals, adjust, and convert back.

Worked examples

Each one is worked to the point where the next move is obvious, not to the answer alone.

1

A recipe serves 4 and calls for 6 ounces of stock. How much stock is needed to serve 10?

  1. 1.Find the rate per serving: 6 / 4 = 1.5 ounces.
  2. 2.Multiply by the new number of servings: 1.5 x 10.

15 ounces

2

A jacket priced at $90 is reduced by 30 percent. What does it cost?

  1. 1.A 30 percent reduction leaves 70 percent of the price.
  2. 2.Multiply rather than subtract twice: 90 x 0.70.

$63

3

Four numbers have a mean of 15. A fifth number is added and the mean becomes 16. What is the fifth number?

  1. 1.The four numbers sum to 15 x 4 = 60.
  2. 2.Five numbers with a mean of 16 sum to 16 x 5 = 80.
  3. 3.The new value is the difference between the two sums.

20

4

A price rises by 10 percent and then falls by 10 percent. What is the overall change?

  1. 1.Percentage changes multiply rather than add, so work with factors.
  2. 2.A rise of 10 percent is a factor of 1.10; a fall of 10 percent is 0.90.
  3. 3.1.10 x 0.90 = 0.99, so the final price is 99 percent of the original.
  4. 4.The fall is taken from a larger amount than the rise was added to, which is why they do not cancel.

A fall of 1 percent

5

Flour is sold as 500 g for 2.40 pounds or 750 g for 3.45 pounds. Which is better value?

  1. 1.Compare on the same basis: price per gram, or per 100 g.
  2. 2.500 g pack: 2.40 / 5 = 0.48 pounds per 100 g.
  3. 3.750 g pack: 3.45 / 7.5 = 0.46 pounds per 100 g.
  4. 4.The larger pack costs less per unit, so it is the better value.

The 750 g pack

6

Of 200 people surveyed, 120 own a car. Of the car owners, 45 also cycle. If a car owner is picked at random, what is the probability they cycle?

  1. 1.Identify the group the question picks from: car owners, not all 200.
  2. 2.That group has 120 people, and 45 of them cycle.
  3. 3.So the probability is 45 / 120 = 3/8.
  4. 4.Dividing by 200 would answer a different question: the chance a surveyed person both drives and cycles.

3/8

Where marks are lost

Each of these has a matching wrong answer waiting among the four choices.

Using the wrong denominator

A question asking for a fraction of one group will offer the fraction of the whole population as a choice. Reread the noun immediately after "of".

Treating successive percentages as additive

Up 10 percent then down 10 percent is a net loss of 1 percent, not a return to the start. The two percentages apply to different amounts.

Confusing mean and median

When a data set has a few extreme values, the mean moves and the median does not. A question about a "typical" value is usually asking for the median.

Answering with the rate instead of the total

Finding 15 ounces per serving is a step, not the answer, when the question asked how much is needed for ten servings.

Adding percentage changes

A 25 percent fall followed by a 40 percent rise is not a 15 percent rise. Percentages are of different amounts, so the factors multiply: 0.75 x 1.40 = 1.05, a 5 percent rise. Work in factors and the trap disappears.

The wrong denominator

A question asking what proportion of the under-40s do something is asking about the under-40s, not about everyone surveyed. Read the noun immediately after 'of' — that noun is the denominator, and the other one will be among the choices.

Because the arithmetic is easy, this area rewards reading more than computation. Slowing down for the sentence that names the quantity is the single highest-return habit available here.

Now work the questions

112 Problem-Solving and Data Analysis questions with a worked explanation for each, on the SAT Math page.

Open SAT Problem-Solving & Data Analysis practice

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SAT® Data Analysis: Method and Practice | IQ Test Center