The words “from June” set the reference count at 72 loans. July adds 90 − 72 = 18 loans.
Divide the increase by the starting count: 18 ÷ 72 = 1/4 = 0.25. The loan count increased by 25%. Adding one quarter of the June count back to the June count reproduces July. Check that 72 × 1.25 = 90.
Compare the other answer choices
20%
The increase is 18 loans, and 18 ÷ 90 = 0.20. That uses the July count as the base. A change from June to July is measured against June.
18%
Subtracting the counts gives 18 loans. That is the size of the increase in loans. To express it as a percent, divide by the starting number of loans first.
125%
The ratio 90 ÷ 72 is 1.25, so July is 125% of June. Subtract the original 100% to isolate the increase.
Which option gives all real solutions of (x − 2)² = 25?
x = −3 or x = 7
x = 7 only
x = −7 or x = 3
x = −5 or x = 5
Answer: x = −3 or x = 7
The entire expression x − 2 is squared. Both 5 and −5 have square 25, so solve x − 2 = 5 and x − 2 = −5.
Adding 2 in each equation gives 7 and −3. Substitute to check: 7 − 2 = 5, and −3 − 2 = −5. Both squares equal 25. Keeping the negative inner value is necessary because squaring erases the sign.
Compare the other answer choices
x = 7 only
Taking x − 2 = 5 gives x = 7. The other branch x − 2 = −5 also squares to 25 and gives a second solution.
x = −7 or x = 3
Subtracting 2 from both 5 and −5 gives 3 and −7. To isolate x in x − 2 = ±5, add 2 to both branches.
x = −5 or x = 5
Both 5 and −5 square to 25. They are possible values of the expression x − 2. The last step shifts each value by 2 to find x.
QUESTION 04Quantitative Reasoning, Algebra, and Statistics (QAS)
Linear equations, slope, and intercepts
A workshop charges a one-time setup fee plus the same price for each panel cut. An order of 3 panels costs $17 and an order of 8 panels costs $32. What is the setup fee?
$8
$3
$17
$14
Answer: $8
Subtract the total prices and the quantities to isolate the variable charge. Five more panels cost $15 more, so each panel costs $3.
The order of three panels contains $9 in variable charges. Subtract $9 from $17 to get the $8 setup fee. Checking the larger order gives 8 + 8 × 3 = 32.
Compare the other answer choices
$3
The cost difference is $15 for five extra panels. The resulting $3 is the price per panel.
$17
The smaller order costs $17 in total. It includes the setup fee and the price of three panels.
$14
Subtracting one panel price from $17 leaves $14. Subtract the cost of all three panels to isolate setup.
Selected Arithmetic, QAS, and AAF foundations for college math placement.
Work through original practice in number sense, quantitative reasoning, and advanced algebra.
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