Student edition of Test 25
The original question order, calculations, and point values are preserved.The conventions below make the allowed inputs explicit.
Lesson 25.7 · Student exercise
Chapter 25 Exercise
Test 25: Defined Operations. Read each new definition from the beginning, preserve the order of its inputs and parentheses, and turn every unfamiliar symbol into familiar arithmetic before solving.
Exercise directions
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Scoring notes
How to read the notation on this page
Here natural numbers mean 0, 1, 2, …; positive integers begin at 1. A denominator must never be zero. In Question 5, use positive integers; in Question 6, the step index n is a positive integer. In Question 7, a is a digit from 1 through 9 and b is a positive integer counting the summands; the last summand has b copies of a.
Reset the symbol. The star and triangle symbols are redefined in different questions. Use only the definition printed directly above the current question.
Keep input order. In a ★ b, the left input replaces a and the right input replaces b. Do not swap them unless the definition proves that you may.
Keep parentheses. Evaluate the innermost custom operation first. A defined operation is not automatically associative.
Read domains. Natural-number, positive-integer, fixed-constant, and denominator restrictions are part of the problem.
Repeated digits. A bar over repeated copies of a digit denotes the numeral formed by writing that digit several times, not multiplication.
Products in Questions 12–13. The expression ab means a × b. In Question 13, the product contains exactly b consecutive factors.
Section I — Fill In
Questions 1–10 · 6 points each · 60 points
Evaluate a nested operation
6 points★
Definition: for the current question, a ★ b=(b+a)×b.
Evaluate (2 ★ 3) ★ 5.
Continue a repeated-digit pattern
6 pointsUse a digit from 1 through 9 as the left input and a positive integer number of summands as the right input. The following three examples illustrate the rule:
6 ★ 2=6+66=72
2 ★ 3=2+22+222=246
1 ★ 4=1+11+111+1111=1234
Using that repeated-digit pattern, evaluate 7 ★ 5.
Pattern convention. A finite list of examples does not logically determine a unique rule. For this original question, use the repeated-digit pattern illustrated consistently by all three examples.
Work backward from a consecutive sum
6 points△
Definition: for natural numbers a,b with b>a,
a △ b=a+(a+1)+(a+2)+⋯+(a+b-1).
If x △ 10=65, find x.
Recover a fixed constant, then evaluate
6 points★
Definition: A is one fixed constant used throughout this question, and
x ★ y= 1xy + 1(x+1)(y+A).
Given 2 ★ 1= 23, find the exact value of 1998 ★ 1999.
Need a hint?
First substitute 2 and 1: 1/2 + 1/[3(1+A)] = 2/3. Subtract 1/2 to find the second fraction, then recover A. For the final sum, use a common denominator or factor out the shared factor 1/1999.
Use GCD and LCM inside a new operation
6 points∧
Definition: a ∧ b=[a,b]+(a,b), where [a,b] is the least common multiple and (a,b) is the greatest common divisor.
Complete both parts.
Need a hint?
For the second part, let g be the GCD and L the LCM. Since L+g=33 and g≥1, L≤32. The unknown x divides L, so x≤32: a search need not continue forever. A shorter route: g divides both 6 and 33, so test g=1 and g=3; L must be a multiple of 6.
Follow a recursive operation
6 points⊙
Definition: a ⊙ 1=a, and for n>1,
a ⊙ n=2[a ⊙ (n-1)]+a.
Given m ⊙ 4=30, complete both parts.
Read a sum of repeated-digit numerals
6 points★
Definition: a ★ b is the sum of the numerals formed by writing the digit a one time, two times, three times, and so on through b times.
For example, the final summand in a ★ b contains exactly b copies of a. If 1 ★ x=123456789, find x.
Keep left and right inputs in order
6 points★
Definition: a ★ b=a×b+a.
The value of x ★ 5 is 100 greater than the value of 5 ★ x. Find x.
Use larger-number and smaller-number selectors
6 points○
Definitions: u ○ v selects the larger of u,v, while u △ v selects the smaller.
Evaluate the exact value of
(0.6 ○ 1726)+(0.625 △ 2333)
(0.3 △ 3499)+(237106 ○ 2.25)
Repeating decimals. Here 0.6 means 0.666…, and 0.3 means 0.333….
Need a hint?
Convert repeating decimals exactly: if t=0.333…, then 10t−t=3, so t=1/3. Doubling gives 0.666…=2/3. Compare positive fractions by cross-multiplication. Make all four selections before adding and dividing.
Recover the fixed parameter m
6 points@
Definition: for natural numbers X,Y,
X @ Y=6XYmX+2Y, where m is one fixed natural number.
Given 1 @ 2=1, find 2 @ 8.
Section II — Extended Response
Questions 11–13 · 20 points each · 60 points
Recover linked parameters in two operations
20 pointsFor numbers x,y, define two new operations:
x ★ y=mx+ny
x △ y=kxy
Here m,n,k are positive integers. Given
1 ★ 2=5(2 ★ 3) △ 4=64
find (1 △ 2) ★ 3.
Use a triangular-number operation
20 points★
Definition: for positive integers a,b,
a ★ b=1+2+3+⋯+ab, where ab means the product a×b.
(1) Find (4 ★ 5)-(2 ★ 3).
(2) If a>b>1 and a ★ b=595, find a and b.
Reverse a product of consecutive factors
20 points★
Definition: for nonzero natural numbers a,b,
a ★ b=a×(a+1)×(a+2)×⋯×(a+b-1).
If (x ★ 3) ★ 2=3660, find x.