DILR guide, section 7

DI table and Venn diagram

These are the two most “mechanical” set types: a missing-data table fills itself if you always attack the cell that is alone in its row or column, and a Venn set falls to one formula plus the extra-memberships idea for max-min questions.

Worked set 5, easy. Attempt it in 15 minutes first.

DI table with missing values

Units sold by four shops from January to March. Some cells are blank.

Shop	Jan	Feb	Mar	Total
W	120	?	150	400
X	?	140	?	420
Y	100	110	?	?
Z	?	?	160	?
Total	450	430	?	?

1X sold 20 more units in March than in January.
2Y’s three-month total equals W’s.
3Z sold 50 more units in January than in February.

Questions

(TITA) How many units did Y sell in March?
Which shop had the highest percentage growth from January to March?
In how many shops did sales fall from January to February?
(TITA) What percentage of the three-month total came in March, to the nearest whole number?

Show solution

Fill one-unknown cells first.

W’s Feb = 400 − 120 − 150 = 130.
X: Jan + Mar = 420 − 140 = 280, and Mar = Jan + 20, so Jan = 130 and Mar = 150.
Y’s total = 400, so Y’s Mar = 400 − 100 − 110 = 190.
Z’s Jan = 450 − (120 + 130 + 100) = 100. Z’s Feb = 430 − (130 + 140 + 110) = 50, which matches the “50 more” clue.
March total = 150 + 150 + 190 + 160 = 650. Grand total = 1,530.

Shop	Jan	Feb	Mar	Total
W	120	130	150	400
X	130	140	150	420
Y	100	110	190	400
Z	100	50	160	310
Total	450	430	650	1,530

Answers. (1) 190. (2) Y, at 90% (W 25%, X 15.4%, Z 60%). (3) 1, namely Z. (4) 42: 650/1,530 = 0.4248, which rounds to 42%. When a TITA answer sits this close to a rounding boundary, compute to three decimal places.

Takeaway. Look for a row or column with exactly one blank; each cell you fill unlocks the next. Use the extra clues only when the totals run out.

12 minutes, timed

Practice set 5: DI table with missing values

Students in two sections, P and Q, each chose one elective. Some cells are blank.

Elective	Section P	Section Q	Total
Finance	24	?	50
Marketing	?	30	?
HR	12	?	?
Operations	?	?	40
Total	80	90	170

1In Section P, Marketing had twice as many students as HR.
2Operations had the same number of students in both sections.

Questions

(TITA) How many Section Q students chose HR?
Which elective was the most popular overall?
(TITA) What percentage of Section P chose Finance?
In how many electives did Section Q have more students than Section P?

Show answers

(1) 14. (2) Marketing, with 54. (3) 30: 24 of 80. (4) 3: Finance, Marketing and HR (Operations is equal). Fill in order: Finance Q = 26; Marketing P = 2 × 12 = 24; Operations P = 80 − 24 − 24 − 12 = 20, so Operations Q is also 20 (total 40, which checks out); HR Q = 90 − 26 − 30 − 20 = 14.

Worked set 6, medium. Attempt it in 15 minutes first.

Venn diagram with max-min

Each of 100 students studies at least one of French, German and Spanish. 50 study French, 40 German and 35 Spanish. 12 study both French and German, 10 both German and Spanish, and 8 both French and Spanish; each pair count includes the students who study all three.

Questions

(TITA) How many study all three languages?
(TITA) How many study exactly one language?
(TITA) How many study only French?
(TITA) Suppose you knew only the three subject totals and that all 100 study at least one language. What is the largest possible number studying all three?

Show solution

100 = 50 + 40 + 35 − (12 + 10 + 8) + x, so x = 5.
Exactly two languages: (12 − 5) + (10 − 5) + (8 − 5) = 15.
Exactly one: 100 − 15 − 5 = 80.
Only French: 50 − 7 (French and German only) − 3 (French and Spanish only) − 5 (all three) = 35.

The max-min idea: extra memberships. The subject totals add up to 125, but there are only 100 students, so 25 memberships are “extra”. A student in exactly two languages accounts for 1 extra; a student in all three accounts for 2. To make the all-three group as large as possible, use 2s: 12 students × 2 = 24, plus one student in exactly two languages for the last 1. The maximum is 12 (and the minimum is 0, using 25 students in exactly two).

Check that 12 is achievable: only French 37, only German 27, only Spanish 23, French and German only 1, all three 12. That makes 100 students, with French 37 + 1 + 12 = 50, German 27 + 1 + 12 = 40 and Spanish 23 + 12 = 35.

Answers. (1) 5. (2) 80. (3) 35. (4) 12.

Takeaway. Fill a Venn diagram from the centre outward. For maximum or minimum overlaps, count the extra memberships and spend them as 1s or 2s.

12 minutes, timed

Practice set 6: Venn diagram

In a survey of 200 people, everyone uses at least one of three apps, X, Y and Z. 110 use X, 90 use Y and 70 use Z. 30 use both X and Y, 25 both Y and Z, and 20 both X and Z; each pair count includes those who use all three.

Questions

(TITA) How many people use all three apps?
(TITA) How many use exactly one app?
(TITA) How many use only Y?
(TITA) If you knew only the three app totals and that everyone uses at least one app, what is the largest possible number using all three?

Show answers

(1) 5: 200 = 110 + 90 + 70 − (30 + 25 + 20) + x. (2) 135: exactly two = 25 + 20 + 15 = 60, so exactly one = 200 − 60 − 5. (3) 40: 90 − 25 − 20 − 5. (4) 35: the totals add to 270, so there are 70 extra memberships; each all-three user takes 2, giving 35. Check: only X 75, only Y 55, only Z 35 and all three 35 make 200.