Patterns & Reasoning

Depth · one concept from first principles · step 2 of 4

Fill a table and complete a line graph from clues

Representative Problem

The line graph shows the number of picture frames a workshop made each month. In July the workshop made 8080 more frames than it made in June, and the number of frames made in July and August together is 640640. Complete the line graph.

(Figure) A line graph titled "Picture Frames Made." The horizontal axis shows the month the count was taken — 44 (April), 55 (May), 66 (June), 77 (July), and 88 (August). The vertical axis shows the number of frames, with major gridlines at 00, 100100, 200200, 300300, and 400400 and each small grid square worth 2020 frames. Points are plotted and joined for April at 280280, May at 380380, and June at 180180; the points for July and August have not yet been plotted.

Picture Frames Made 0 100 200 300 400 (frames) 4 5 6 7 8 Month
Measurement & data
Your answer
How to solve
Strategy Identify Subproblems — I solve in order from what I can already know: first July (June plus 80), then August (the 640 total minus July). Each is a one-step subproblem, and finding August uses the total backwards.
1STEP 1

Find July

July is 80 more than June, and June is 180, so July = 180 + 80 = 260.

180 + 80 = 260
2STEP 2

Find August

July and August together make 640, so August = 640 - 260 = 380.

640 - 260 = 380
3STEP 3

Plot the points

On the graph (each square = 20), July 260 sits 3 squares above 200, August 380 sits 4 squares above 300; join the points in order.

260 = 200 + 3× 20, 380 = 300 + 4× 20
Answer
July = 260 frames, August = 380 frames
Check the clues: July 260 is indeed 80 more than June 180, and July 260 + August 380 = 640 as required. Both values land neatly on grid lines (260 and 380 are multiples of 20), so the plotted points are valid.
Takeaway

Fill in what you can figure out first, then use the total to get the rest - then just plot!

  • Find July
  • Find August
  • Plot the points
Where next?
Another one like thissuggested

A parent dashboard for the family lives at sensimlab.com.

▶ Practice — 8 problems

The same shape shows up later on the AMC, as Branching State Enumeration, Lattice / Grid Pattern from Small Cases, Small-Domain Constrained Search. Why the shapes repeat →