Reasoning · Grade 6-1 Ratio and Rate (2)
Problem
Write the relation in all three directions.
A batting average is a part-to-whole rate like a price per item, so knowing any two of the three numbers pins down the third.
6.RP.A.1Make A Systematic ListThe one relation between hits, at-bats and average can be read in all three directions, so any two of them give the third.
An average is a fixed rate of hits per at-bat, so the same rate turns any at-bat count into a hit count.
Dividing undoes multiplying, so the same relation can be run forwards to find hits or backwards to find at-bats.
Hits over at-bats gives 0.316.
The fraction 12/38 just means 12 shared among 38, and the division carried a few places past the decimal point gives as many digits as the rounding rule needs.
5.NF.B.3Make A Systematic ListAt-bats times average gives 11 hits.
An average of 0.25 means one hit in every four at-bats, so counting how many fours are inside 44 gives the hits without any decimal work.
6.RP.A.3Work BackwardsHits over average gives 35 at-bats.
Dividing by 0.4 asks how many groups of four tenths fit into 14, and the answer must come out larger than 14 because at-bats always outnumber hits.
6.NS.B.3Work BackwardsAdding today's game gives 0.292.
Totals are what accumulate over a season, so a new game is folded in by adding to the top and the bottom of the same fraction — the same way the problem's own example turns Liam's 38 and 12 into 43 and 15.
6.RP.A.3Make A Systematic ListA 1-for-3 game lifts two and lowers one.
Checking every player the same way, with the same 1 hit and 3 at-bats added, is what keeps a 'who improves' question from turning into a guess.
6.RP.A.3Make A Systematic ListOnly players below 1 in 3 improve.
A new stretch of play acts like a weight pulling the season rate toward the rate of that stretch, which is why one comparison against 0.333 decides every player at once.
6.RP.A.3Look For A PatternA new game pulls your average toward the rate you hit at that day — 1 for 3 is 0.333, so it lifts everyone batting under 0.333 and lowers everyone above it.