Translate a rate and honor whole-package constraints
Arithmetic Reasoning asks you to model a word problem, not just identify a convenient operation. Start by writing what the final answer measures. A number of fasteners, a number of packages and a number of completed kits are different quantities. A proportion is useful only when its units are arranged consistently; reversing one ratio changes the relationship. Worked example: a workshop needs 8 kits with 9 clips in each. The total requirement is 8 kits × 9 clips/kit = 72 clips. Clips come in sealed packages of 20. Dividing 72 clips by 20 clips/package gives 3.6 packages as a capacity calculation. Because only whole packages can be bought, four are needed. They hold 80 clips and leave 8 unused. Three packages hold only 60, so ordinary nearest-integer rounding would fail the requirement. The rounding decision belongs at the end of the model. If the task instead allowed a measured length of ribbon to be cut from a roll, a fractional number of meters might be valid. If packages could not be shared across kits, the purchase calculation would also change. Do not silently add either permission or restriction. Our diagnostic expressly allows unused fasteners to be shared and forbids splitting a package at purchase. Check a proposed answer by reconstructing its capacity. A feasible quantity may still exceed the requested minimum. A smaller quantity may be close numerically but leave a shortage. Writing both capacity and surplus or shortage distinguishes those errors. This is ordinary academic allocation arithmetic; it is not instruction for operating tools or carrying out a military task.
A volunteer workshop needs 9 identical kits, each requiring 14 fasteners. Fasteners are sold only in sealed packages of 20, and unused fasteners can be shared across kits. There is no other stock. What is the minimum number of packages needed?
- The kits need 9 × 14 = 126 fasteners. Six packages contain 120, which is insufficient. Seven contain 140, enough with 14 left over. Because packages cannot be split at purchase, the quotient 126/20 = 6.3 must be rounded up, not to the nearest whole number.
Compare the other answer choices
Eight packages are sufficient but exceed the minimum by one package.
One package per kit ignores the statement that unused fasteners can be shared.
Six packages leave a shortage of 6 fasteners; rounding 6.3 down does not meet the requirement.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which evidence or rule supports your decision and why the alternatives do not fit.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ASVAB SubtestsSolve Proportions and their Applications