Data Representation · Lesson 6

Quick arithmetic from data

Some questions need a quick calculation with values you read from a figure: a difference, a ratio, about how many times as much, or a percent change. No calculator is allowed, and none is needed. The numbers are chosen to work out cleanly, and the choices are usually far enough apart that a rounded estimate picks the right one.

Round first, then compute. If the values are 198 and 401, think 200 and 400. How many times as much means divide: 100 g is 4 times 25 g. How many more means subtract: 100 g is 75 g more than 25 g. Mixing up those two is a common wrong choice.

Percent change = change ÷ starting value × 100%. Going from 50 g to 70 g is a change of 20 g, and 20 ÷ 50 = 0.4, so the increase is 40%. The starting value is the one the question says the change is from. Before you commit, compare your answer with the spread of the choices: if two choices are close, compute exactly; if they are far apart, your estimate is enough.

How Much Salt Dissolves

Students measured the solubility of two salts, Salt P and Salt Q: the greatest mass of each salt, in grams (g), that dissolves in 100 g of water. They tested each salt at five water temperatures.

Table 1
Water temperature (°C)Salt P solubility (g per 100 g water)Salt Q solubility (g per 100 g water)
01240
202042
403244
604846
806048
Solubility in grams per 100 g of water at 0, 20, 40, 60 and 80°C. Salt P: 12, 20, 32, 48 and 60 g, rising at every step. Salt Q: 40, 42, 44, 46 and 48 g, rising by 2 g for each 20°C.

Worked example Easy

Show the whole passage

How Much Salt Dissolves

Students measured the solubility of two salts, Salt P and Salt Q: the greatest mass of each salt, in grams (g), that dissolves in 100 g of water. They tested each salt at five water temperatures.

Table 1
Water temperature (°C)Salt P solubility (g per 100 g water)Salt Q solubility (g per 100 g water)
01240
202042
403244
604846
806048
Solubility in grams per 100 g of water at 0, 20, 40, 60 and 80°C. Salt P: 12, 20, 32, 48 and 60 g, rising at every step. Salt Q: 40, 42, 44, 46 and 48 g, rising by 2 g for each 20°C.

According to Table 1, at 20°C, how many more grams of Salt Q than of Salt P dissolve in 100 g of water?

  1. 20 g
  2. 22 gAnswer
  3. 42 g
  4. 62 g

How to solve it

  1. Find the 20°C row of Table 1: Salt P 20 g, Salt Q 42 g.
  2. How many more means subtract: 42 g − 20 g = 22 g.

Why each choice is right or wrong

  • A. Incorrect. 20 g is Salt P’s solubility at 20°C, one of the two values, not their difference.
  • B. Correct. In the 20°C row of Table 1, Salt Q’s solubility is 42 g and Salt P’s is 20 g, and 42 g − 20 g = 22 g.
  • C. Incorrect. 42 g is Salt Q’s solubility at 20°C, one of the two values, not their difference.
  • D. Incorrect. 62 g adds the two values (42 g + 20 g). How many more asks for a difference, so subtract.

Worked example Medium

Show the whole passage

How Much Salt Dissolves

Students measured the solubility of two salts, Salt P and Salt Q: the greatest mass of each salt, in grams (g), that dissolves in 100 g of water. They tested each salt at five water temperatures.

Table 1
Water temperature (°C)Salt P solubility (g per 100 g water)Salt Q solubility (g per 100 g water)
01240
202042
403244
604846
806048
Solubility in grams per 100 g of water at 0, 20, 40, 60 and 80°C. Salt P: 12, 20, 32, 48 and 60 g, rising at every step. Salt Q: 40, 42, 44, 46 and 48 g, rising by 2 g for each 20°C.

According to Table 1, about how many times as much Salt P dissolves in 100 g of water at 80°C as at 20°C?

  1. 2 times
  2. 3 timesAnswer
  3. 5 times
  4. 40 times

How to solve it

  1. Read Salt P in two rows of Table 1: 60 g at 80°C and 20 g at 20°C.
  2. How many times as much means divide: 60 ÷ 20 = 3.
  3. Check the spread: the choices 2, 3, 5 and 40 are far enough apart that the exact quotient settles it.

Why each choice is right or wrong

  • A. Incorrect. 2 times compares 80°C (60 g) with 40°C (32 g), the wrong starting row; 60 ÷ 32 is about 2.
  • B. Correct. Salt P’s solubility is 60 g at 80°C and 20 g at 20°C, and 60 ÷ 20 = 3, so 3 times as much dissolves.
  • C. Incorrect. 5 times compares 80°C (60 g) with 0°C (12 g), the wrong starting row.
  • D. Incorrect. 40 is the difference, 60 g − 20 g = 40 g. How many times as much asks you to divide, not subtract.

Trap Percent of the wrong base

The most common slip in a percent change is dividing by the new value instead of the starting value. From 80 g to 100 g, the change is 20 g, and 20 ÷ 80 = 25%; dividing by 100 instead gives 20%, which is wrong. Divide by the value the change starts from. The wrong-base answer is usually one of the choices, so finding your number among the choices doesn’t prove you used the right base.