This is an excellent item for testing analytical thinking and has been used in a variety of different guises for decades. While there are different ways to approach this question, a common form of reasoning proceeds as follows:
• If it takes 5 minutes for 5 machines to make 5 widgets, then it will take 5 minutes for a machine to make a widget.
• Since a machine can make a widget in 5 minutes, then 100 machines would make 100 widgets in 5 minutes. Answer: 5 minutes
Now let’s look at how powerful this format can be for testing reasoning skills at higher levels. The following excerpt is taken from my recent book Intelligence: Where We Were, Where we are and Where We’re Going.
One of the most comprehensive studies of mathematical giftedness was conducted in the Soviet Union between 1955 and 1966, by a team of researchers headed by psychologist Vadim Krutetskii. In his attempts to determine how the mathematically gifted differ from others, Krutetskii and his research team presented a series of mathematics problems to students in grades 6 through 8 (ages 11 to 13) and observed their thinking processes.
One of the problems that Krutetskii and his team used to measure this aptitude was the following:
Problem: A factory is expected to turn out x tools over a definite period, and therefore, planned to make y tools per day. The workers exceeded the quota and each day made ztools more than was planned. How many days before the projected deadline did the plant fill its order?
Any student who was able to solve this problem on first try would be assessed as capable and moved onward to more challenging problems. A student who was unsuccessful, would be given variant 1 of the problem (see below) and on successful completion moved to the next higher variant until ultimately reaching the abstract version given above. The rate at which a student progressed through this sequence from concrete to abstract was used to assess their ability to generalize from specific to general problems.
Variant 1: (numbers only) A factory is expected to turn out 100 tools over a definite period, and therefore, planned to make 4 tools per day. The workers exceeded the quota and each day made one more tool than was planned. How many days before the projected deadline did the plant fill its order?
[Answer: 5 days]
Variant 2: (1 variable) A factory is expected to turn out x tools over a definite period, and therefore, planned to make 4 tools per day. The workers exceeded the quota and each day made one more tool than was planned. How many days before the projected deadline did the plant fill its order?
[Answer: x/4 – x/5 days]
Variant 3: (2 variables) A factory is expected to turn out x tools over a definite period, and therefore, planned to make y tools per day. The workers exceeded the quota and each day made one more tool than was planned. How many days before the projected deadline did the plant fill its order?
[Answer: x/y – x/(y + 1) days]
A student moving to the fourth variant, i.e., the original problem, would be expected to discover the answer x/y – x/(y + z).
In describing the results of this research, Krutetskii noted innate differences in the cognitive capacities of children, while carefully sidestepping the Soviet prohibition on linking talent to inheritance:
The difference between capable, average, and incapable pupils, as our research permits us to conclude, comes down to the following. In able pupils these associations can be formed “on the spot”; in this sense they are “born,” if one can so express it, already generalized, with a minimal number of exercises. In average pupils these associations are established and reinforced gradually, as a result of a whole series of exercises. They form isolated, concrete associations, related only to a given problem, “on the spot.” Through single-type exercises these associations are gradually transformed into generalized associations. In incapable pupils, even the isolated, concrete associations are formed with difficulty, their generalizations are still more difficult, and sometimes such generalizations do not occur at all.
